Interior decay of solutions to elliptic equations with

Interior decay of solutions to elliptic equations with respect to frequencies at the boundary Michele Di Cristo Luca Rondiy ... Moreover, in the real-...

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Interior decay of solutions to elliptic equations with respect to frequencies at the boundary Michele Di Cristo∗

Luca Rondi†

Abstract We prove decay estimates in the interior for solutions to elliptic equations in divergence form with Lipschitz continuous coefficients. The estimates explicitly depend on the distance from the boundary and on suitable notions of frequency of the Dirichlet boundary datum. We show that, as the frequency at the boundary grows, the square of a suitable norm of the solution in a compact subset of the domain decays in an inversely proportional manner with respect to the corresponding frequency. Under Lipschitz regularity assumptions, these estimates are essentially optimal and they have important consequences for the choice of optimal measurements for corresponding inverse boundary value problem. AMS 2010 Mathematics Subject Classification Primary 35B30. Secondary 35J25, 35R01, 35R30. Keywords elliptic equations, interior decay, frequency method, Riemannian manifolds, Steklov eigenfunctions.

1

Introduction

An important motivation for our study comes from elliptic inverse boundary value problems, such as the Calder´ on problem. Let us consider a bounded domain Ω ⊂ RN , N ≥ 2, which is regular enough and let γ be a positive bounded function which is bounded away from zero and that corresponds to the background conductivity of a conducting body contained in Ω. The aim of the inverse problem is to recover perturbations of the background conductivity, for example inhomogeneities, by performing suitable electrostatic measurements at the boundary of current and voltage type. Such a problem comes from several types of nondesctructive evaluation problems in materials, where the aim is to detect the presence of flaws, as well as from medical imaging problems, where the aim is to detect the presence of tumors. Namely, if γ˜ is the perturbed conductivity, one usually prescribes the voltage f on the boundary of Ω and measures the corresponding current still on the boundary, that is, γ˜ ∇˜ u ·ν on ∂Ω, where ν is the outer normal on ∂Ω and u ˜, the electrostatic potential in Ω, is the solution to the Dirichlet boundary value problem  div(˜ γ ∇˜ u) = 0 in Ω (1.1) u ˜=f on ∂Ω. ∗

Dipartimento di Matematica, Politecnico di Milano, Italy. E-mail: [email protected] † Dipartimento di Matematica, Universit` a degli Studi di Milano, Italy. E-mail: [email protected]

1

By changing the Dirichlet datum f , one can perform two or more measurements. One often assumes that the perturbation is well contained inside Ω, that is, γ˜ coincides with the ˜ a known background conductivity γ in a known neighbourhood of ∂Ω, that is, outside Ω, open set compactly contained in Ω. Since [16], it has been clear that one source of instability for elliptic inverse boundary value problems is due to the interior decay of solutions. We consider the solution u to the Dirichlet problem in the unperturbed body, that is, with γ˜ replaced by the background conductivity γ, namely  div(γ∇u) = 0 in Ω (1.2) u=f on ∂Ω. The possibility to recover stably information on the unknown perturbation, using the additional measurement depending on the Dirichlet datum f , is directly related to the decay of ˜ the region where the perturu, or ∇u, in the interior of the domain Ω, in particular in Ω, bation may be present. Therefore it is particularly important to establish decay properties ˜ depending on the Dirichlet datum f and the distance of Ω ˜ from ∂Ω, which is of u in Ω, exactly the issue we address in this paper. In [16], it was assumed that the domain Ω is B1 , a ball of radius 1, and the background conductivity is homogeneous, γ ≡ 1. Then it was shown that the exponential instability of the inverse problem of Calder´ on is due to the fact that solutions to (1.2) with boundary values f given by spherical harmonics decay in the interior exponentially with respect to the degree of the spherical harmonic itself. We note that a spherical harmonic is a Steklov eigenfunction for the Laplacian on B1 with corresponding Steklov eigenvalue given by its degree. We also note that the degree is exactly equal to the frequency as defined in Definition 2.8. The Weyl law on the asymptotic behaviour of Steklov eigenvalues is the other key ingredient. We recall that µ is a Steklov eigenvalue and φ|∂Ω is its corresponding Steklov eigenfunction if φ is a nontrivial solution to  div(γ∇φ) = 0 in Ω (1.3) γ∇φ · ν = µφ on ∂Ω. We also recall that, for f ∈ H 1/2 (∂Ω), f 6= 0, we call its frequency the number frequency(f ) =

|f |2H 1/2 (∂Ω) kf k2L2 (∂Ω)

.

We refer to Definition 2.8 for a precise statement, here we just note that, if φ|∂Ω is the trace of a nontrivial solution to (1.3), then its frequency is essentially proportional to the Steklov eigenvalue µ. The ideas of [16] have been generalised to other elliptic boundary value and scattering inverse problems in [5, 6] and to the parabolic case in [7], showing that exponential instability unfortunately holds in all these cases. Besides showing the instability nature of these problems, these results provide hints on the choice of optimal measurements, where optimality may be in the sense of distinguishability as defined in [15], see also [12]. As suggested in these papers, if one has at disposal a fixed and finite number n of measurements, one should choose the first n eigenfunctions of a suitable eigenvalue problem involving the perturbed conductivity γ˜ and the background 2

one γ. Since the conductivity γ˜ is unknown, by using the arguments developed in [16] the best choice should be to employ the first n spherical harmonics, at least when the domain is a ball and the background conductivity is constant. In a general case, it seems reasonable to assume that the correct replacement for spherical harmonics is given by Steklov eigenfunctions. Indeed, the interior decay of the solution corresponding to a Steklov eigenfunction is very fast with respect to the Steklov eigenvalue, at least in a smooth case. For example, in [14] it is shown that when ∂Ω is C ∞ and γ is C ∞ the decay is faster than any power. Moreover, in the real-analytic case, the decay is still of exponential type, as shown first for surfaces in [18] and then for higher dimensional manifolds in [8]. Consequently, the information carried by the measurements corresponding to boundary data given by high order Steklov eigenfunctions rapidly degrades in the interior of the domain, thus it is of little help for the reconstruction of perturbations of the background conductivity far from the boundary. However, from these examples it seems that the worst case scenario is when the domain and the background conductivity are real-analyitc, because in this case the interior decay of the solution corresponding to a Steklov eigenfunction is indeed of exponential type with respect to the Steklov eigenvalue, or when the domain and the background conductivity are smooth, say C ∞ , since the interior decay is still very fast in this case. Here, instead, we are interested in understanding the interior decay when the domain and the coefficients are not particularly smooth and also when f is not a Steklov eigenfunction. In fact, in some occasions it might be very difficult to employ a Steklov eigenfunction and we wish to show that the decay might be actually due just to the frequency of the boundary datum f , without the much stronger assumption that f is a Steklov eigenfunction. For example, this might be significant for the choice of optimal measurements in a partial data scenario, that is, when data are assigned and collected only on a given portion of the boundary. We also wish to mention that, for the Calder´on problem, the dependence of the dis˜ from the boundary of Ω, rather than on the choice of tinguishability on the distance of Ω boundary measurements, has been carefully analysed in [10, 1] in two dimensions and in [9] in higher dimensions. Our decay estimate, since the dependence on such a distance is explicitly given, may also be of interest in this kind of analysis of the instability. We describe the main estimates we are able to prove, Theorems 4.1 and 4.2. We assume that Ω is a C 1,1 domain and that γ is Lipschitz continuous. We can also assume that γ is a symmetric conductivity tensor, and not just a scalar conductivity, or that the underlying metric in Ω is not the Euclidean one but a Lipschitz Riemannian one. We call Φ the frequency of the Dirichlet boundary datum f ∈ H 1/2 (∂Ω), f 6= 0. Whenever f has zero mean on ∂Ω, we may use another notion of frequency, which we call lower frequency and which is given by kf k2L2 (∂Ω) lowefrequency(f ) = . kf k2H −1/2 (∂Ω) We refer to Definition 2.9 for a precise statement. Here we point out that, if we call Φ1 the lower frequency of f , then Φ1 ≤ Φ. On the other hand, if φ|∂Ω is the trace of a nontrivial solution to (1.3) with µ > 0, then also its lower frequency is essentially proportional to the Steklov eigenvalue µ.

3

For d > 0 small enough, we call Ωd the set Ωd = {x ∈ Ω : dist(x, ∂Ω) > d}. The first result is the following. We can find two positive constants C1 and C2 , depending on Ω, the Riemannian metric on it, and the coefficient γ, such that if dΦ ≥ C1 , then the function u solving (1.2) satisfies R Z k∇uk2 2 (1.4) k∇uk ≤ C2 Ω . dΦ Ωd We refer to Section 4, and in particular to Theorem 4.1, for the precise statement. If we are interested in the decay of u instead of its gradient, when f has zero mean on ∂Ω, we obtain an analogous result but we need to replace the frequency Φ with the lower frequency Φ1 . Namely, we can find two positive constants C1 and C2 , depending on Ω, the Riemannian metric on it, and the coefficient γ, such that if dΦ1 ≥ C1 , then the function u solving (1.2) satisfies R Z u2 dσ 2 . (1.5) u dσ ≤ C2 ∂Ω dΦ1 ∂Ωd We refer to Section 4, and in particular to Theorem 4.2, for the precise statement. As an easy consequence of (1.5), under the same assumptions we obtain that, for two positive constants C1 and C2 , depending on Ω, the Riemannian metric on it, and the coefficient γ, if dΦ1 ≥ C1 , then the function u solving (1.2) satisfies R Z k∇uk2 2 (1.6) k∇uk ≤ C2 Ω 2 . d ΦΦ1 Ωd See Corollary 4.3 for the precise statement. We conclude that, if f = φ|∂Ω is the trace of a nontrivial solution to (1.3) with µ > 0, then, possibly with different constants C1 and C2 , if dµ ≥ C1 , then the function φ solving (1.3) satisfies R Z k∇φk2 2 (1.7) k∇φk ≤ C2 Ω 2 2 , d µ Ωd see Remark 4.4 for a precise statement. Let us briefly comment on the difference between these estimates. Assuming that f ∈ H 1/2 (∂Ω), f 6= 0 with zero mean on ∂Ω, since Φ1 ≤ Φ, we have that D in general decays faster than H. Actually, by Corollary 4.3, we have that, as Φ1 grows, D decays like Φ−1 Φ1−1 , −1 that is, at least like Φ−2 1 , whereas H decays like Φ1 . Moreover, if f coincides with a Steklov eigenfunction with Steklov eigenvalue µ > 0, then, up to a constant, Φ, Φ1 and µ are of the same order, therefore for Steklov eigenfunctions we obtain a decay of order µ−2 , a result which is in accord with the estimate one can prove using the technique of [14]. In fact, an indication of the optimality of our decay estimates comes from the analysis developed in [14] when f = φ|∂Ω is a Steklov eigenfunction, with positive Steklov eigenvalue µ. Following the idea of the proof of [14, Theorem 1.1], it is evident that one can estimate u(x), for any x ∈ Ωd , by a constant times µ−1 kf kH 1/2 (∂Ω) provided the Green’s function Gγ (x, ·) satisfies (1.8)

kΛγ (Gγ (x, ·))kH 1/2 (∂Ω) , kγ∇Gγ (x, ·) · νkH 1/2 (∂Ω) ≤ C˜ 4

where Λγ is the so-called Dirichlet-to-Neumann map. Roughly speaking, (1.8) corresponds to an H 2 -bound of Gγ (x, ·) away from x, which is what one obtains assuming the conductivity √ √ γ is Lipschitz continuous. Since kf kH 1/2 (∂Ω) is of the order of µkf kL2 (∂Ω) = µkukL2 (∂Ω) and, as we already pointed out, the frequency and the lower frequency of f are of the same order of µ, one can obtain an estimate that is perfectly comparable with (1.5). If one wishes to prove a decay of higher order, like u(x) bounded by a constant times µ−2 kf kH 1/2 (∂Ω) , by the same technique of [14], one should estimate the functions appearing in (1.8) in terms of the H 3/2 (∂Ω) norm instead of the H 1/2 (∂Ω) norm, which corresponds to an H 3 -bound of Gγ (x, ·) away from x. Usually Lipschitz regularity of γ is not enough to infer H 3 -bounds, something like C 1,1 regularity would be required instead, therefore, under our weak regularity assumptions, our estimate (1.5) seems to be optimal even for Steklov eigenfunctions. Another indication of the optimality of our decay estimates comes from the analysis developed in [3]. In [3] the authors introduce the so-called penetration function and study its properties for two dimensional domains, in particular for the two dimensional unit ball. They are particularly interested in low regularity cases, thus they allow discontinuous conductivity tensors. Their aim is to obtain estimates in homogenisation theory, but their results can be easily interpreted as distinguishability estimates with a finite number of boundary measurements for corresponding inverse boundary value problems. In particular, using their notation, if Vn is the space of trigonometric polynomials of degree n on ∂B1 (0) ⊂ R2 , d ∈ (0, 1) is a constant, and A = γ is a symmetric conductivity tensor which is Lipschitz continuous, we can show that the penetration function Ξ(Vn , d) satisfies, for a suitable constant C, (1.9)

Ξ(Vn , d) ≤ C(dn)−1 .

In fact, for any f which is orthogonal to Vn in L2 (∂Ω), we have that its frequency Φ is at least n + 1 and also its lower frequency Φ1 is at least n + 1. Therefore (1.9) directly follows from (1.6). Such a result considerably improves the estimate of [3, Theorem 3.4], which is however valid for a wider class of conductivity tensors including discontinuous ones. Moreover, they give evidence by some explicit examples that, when discontinuous conductivity tensors are allowed, a lower bound for the penetration function is of order n−1/2 . It would be interesting to match such a lower bound by an estimate like (1.4) when γ is discontinuous, but such an estimate would require a completely different method from the one used here. About the technique we developed to obtain our estimates, let us begin by considering (1.4), where we use an ordinary differential equation argument that allows us to estimate the decay of Z γk∇uk2

D(d) = Ωd

when d is positive, and small enough. We closely follow the so-called frequency method introduced in [11] to determine unique continuation properties of solutions to elliptic partial differential equations. In [11], the local behaviour, near a point x0 ∈ Ω, of a solution u to div(γ∇u) = 0 in Ω was analysed, even in the case of a symmetric conductivity tensor γ. A key point of the method was to reduce, locally near x0 , the elliptic equation with a symmetric conductivity tensor to an equation in a special Riemannian manifold with a scalar conductivity. By a special Riemannian manifold we mean one whose metric can be written in a special form in terms of polar coordinates centred at x0 . Such a reduction is made possible by the technique developed in [2]. 5

Here we need to perform a similar construction, the only difference, and the main novelty, is that instead of considering a local modification near a point we consider a global one near the boundary of the domain. Indeed, in order to develop our analysis, we need that ∂Ωd depends on d smoothly enough or, equivalently, that the distance function from the boundary is smooth enough, say C 1,1 , in a neighbourhood of the boundary. By [4], see Theorem 2.4, this is true in the Euclidean setting provided ∂Ω is C 1,1 as well. In the Riemannian setting a similar result is much harder to prove. On the other hand, by exploiting the technique of [2] and suitably changing the metric near the boundary, we can reduce to the case where the distance from the boundary, in the Riemannian metric, is smooth enough since it coincides with the distance from the boundary in the Euclidean metric in a neighbourhood of ∂Ω. We believe that such a construction, besides being crucial for the proof of our decay estimates, is of independent interest and is one of the major achievement of the paper. The major part of the construction is contained in Proposition 3.5 and Theorem 3.9, with one interesting application developed in Proposition 3.7. Our argument is based on the notion of frequency, which we essentially take from [11], and which is given by Z D(d) N (d) = where H(d) = γu2 dσ. H(d) d ∂Ω We note that N (0) is of the same order of the frequency of the boundary datum f . We need to compute the derivative of D and of H, a task we perform following the analogous computations of [11]. In particular, for D0 (d) we use the coarea formula and a suitable version of the Rellich identity which is given in Lemma 4.5. Instead, we compute H 0 (d) by a straightforward application of Proposition 3.7. The proof of (1.5) follows analogous lines of that of (1.4) by replacing D with H and H with Z E(d) = γu2 . Ωd

However there are some additional technical difficulties to be taken care of, see the proof of Theorem 4.2 in Section 4. Moreover, the crucial link between the quotient H(0)/E(0), which plays the role of N (0), and the lower frequency Φ1 is provided by the estimate of Proposition 2.17. The plan of the paper is as follows. In Section 2 we present the preliminary results that are needed for our analysis. In particular, we first discuss the regularity of domains and of the corresponding distance from the boundary, with the main result here being Theorem 2.4 which is taken from [4]. We also give the precise definitions of frequencies we use. Then we review the Riemannian setting and the Dirichlet and Neumann problems for elliptic equations in the Euclidean and in the Riemannian setting, pointing out what happens if one suitably changes the underlying metric, see Remarks 2.12 and 2.13. For instance, Remark 2.13 allows us to pass from a symmetric conductivity tensor in the Euclidean setting to a scalar conductivity in the Riemannian one. We also briefly discuss Steklov eigenvalues and eigenfunctions. In Section 3, we investigate the distance function from the boundary in the Riemannian setting. Here the crucial result is Proposition 3.5 which, together with Theorem 3.9 and Remark 2.12, allows us to assume, without loss of generality, that the distance function from the boundary in the Riemannian case has the same regularity as in 6

the Euclidean case. Another important technical result in this section is Proposition 3.7. Finally, in Section 4, we state and prove our main results, the decay estimates contained in Theorems 4.1 and 4.2 and Corollary 4.3. Acknowledgements The authors are partly supported by GNAMPA, INdAM, through 2018 and 2019 projects. The authors wish to thank Eric Bonnetier for pointing them out reference [3].

2

Preliminaries

Throughout the paper the integer N ≥ 2 will denote the space dimension. For any (column) vectors v, w ∈ RN , hv, wi = v T w denotes the usual scalar product on RN . Here, and in the sequel, for any matrix A, AT denotes its transpose. For any x = (x1 , . . . , xN ) ∈ RN , we denote x = (x0 , xN ) ∈ RN −1 × R. We let ei , i = 1, . . . , N , be the vectors of the canonical base and we call π 0 the projection onto the first (N − 1) components and πN the projection onto the last one, namely, for any x ∈ RN , π 0 (x) = x0 = (x1 , . . . , xN −1 )

and πN (x) = xN .

For any s > 0 and any x ∈ RN , Bs (x) denotes the open ball contained in RN with radius s and center x, whereas Bs0 (x0 ) denotes the open ball contained in RN −1 with radius s and S 0 N center x . Finally, for any E ⊂ R , we denote Bs (E) = x∈E Bs (x). For any Borel E ⊂ RN N ×N (R) the space of real-valued N × N symmetric matrices we let |E| = LN (E). We call Msym and by IN we denote the identity N × N matrix. We recall that we drop the dependence of any constant from the space dimension N .

2.1

Regular domains and the distance from the boundary

Definition 2.1 Let Ω ⊂ RN be a bounded open set. Let k be a nonnegative integer and 0 ≤ α ≤ 1. We say that Ω is of class C k,α if for any x ∈ ∂Ω there exist a C k,α function φx : RN −1 → R and a neighbourhood Ux of x such that for any y ∈ Ux we have, up to a rigid transformation depending on x, y = (y 0 , yN ) ∈ Ω if and only if yN < φx (y 0 ). We also say that Ω is of class C k,α with positive constants r and L if for any x ∈ ∂Ω we can choose Ux = Br (x) and φx such that kφx kC k,α (RN −1 ) ≤ L. Remark 2.2 If Ω ⊂ RN , a bounded open set, is of class C k,α then there exist positive constants r and L such that Ω is of class C k,α with constants r and L with the further condition, when k ≥ 1, that for any x ∈ ∂Ω we have ∇φx (x0 ) = 0. We note that a bounded open set of class C 0,1 is said to be of Lipschitz class and that typically one assumes at least that k + α ≥ 1. Definition 2.3 Let Ω ⊂ RN be a bounded open set. For any x ∈ RN , its distance from the boundary of Ω is dist(x, ∂Ω) = inf kx − yk = min kx − yk. y∈∂Ω

y∈∂Ω

7

We call ϕ : RN → R the signed distance function from the boundary of Ω as follows. For any x ∈ RN  dist(x, ∂Ω) if x ∈ Ω, ϕ(x) = −dist(x, ∂Ω) otherwise. We call, for any d ∈ R, Ωd = {x ∈ RN : ϕ(x) > d}

and ∂Ωd = {x ∈ RN : ϕ(x) = d}.

Finally, for any d > 0, we call U d = {x ∈ Ω : ϕ(x) < d}. The regularity of the signed distance function from the boundary has been thoroughly investigated in [4]. Here we are interested in particular in the case of bounded open sets of class C 1,1 which is treated in [4, Theorem 5.7]. Namely the following result holds true. Theorem 2.4 Let us fix positive constants R, r and L. Let Ω ⊂ BR (0) ⊂ RN be a bounded open set of class C 1,1 with constants r and L. Then there exists d˜0 > 0, depending on r and L only, such that, if we call U = {x ∈ RN : |ϕ(x)| < d˜0 }, for any x ∈ U there exists a unique y = P∂Ω (x) ∈ ∂Ω such that kx − P∂Ω (x)k = dist(x, ∂Ω). Moreover, ϕ is differentiable everywhere in U and we have (2.1)

(∇ϕ(x))T = −ν(P∂Ω (x)) for any x ∈ U,

where ν denotes the exterior normal to Ω, which we assume to be a column vector. In particular, k∇ϕk = 1 in U. Finally, we have that P∂Ω ∈ C 0,1 (U ), with C 0,1 norm bounded by r, L and R only, and, through (2.1), we also have that ϕ ∈ C 1,1 (U ), with C 1,1 norm bounded by r, L and R only. Proof. It easily follows by using the arguments of the proof of [4, Theorem 5.7].  Let us note that, under the assumptions of Theorem 2.4, for any 0 ≤ |d| < d˜0 , we have that Ωd is a bounded open set of class C 1,1 and ∂(Ωd ) = ∂Ωd . Moreover, for any x ∈ ∂(Ωd ), if ν(x) denotes the exterior normal to Ωd , then (∇ϕ(x))T = −ν(x) = −ν(P∂Ω (x)). ×N Definition 2.5 Let Ω ⊂ RN be a bounded open set. We say that A = A(x) ∈ MN sym (R), ×N x ∈ Ω, is a symmetric tensor in Ω if A ∈ L∞ (Ω, MN sym (R)). ×N We say that a symmetric tensor A in Ω is Lipschitz if A ∈ C 0,1 (Ω, MN sym (R)) and that a symmetric tensor A in Ω is uniformly elliptic with constant λ, 0 < λ < 1, if

λkξk2 ≤ hA(x)ξ, ξi ≤ λ−1 kξk2

for almost any x ∈ Ω and any ξ ∈ RN .

If Ω is of class C 1,1 and A is a Lipschitz conductivity tensor, we can extend A outside Ω keeping it Lipschitz, and, in case, uniformly elliptic as well. Namely we have. 8

Proposition 2.6 Let us fix positive constants R, r and L. Let Ω ⊂ BR (0) ⊂ RN be a bounded open set of class C 1,1 with constants r and L. Let A be a Lipschitz symmetric tensor in Ω. Then there exists a Lipschitz symmetric tensor A˜ in RN such that A˜ = A in Ω

and

A˜ = IN outside BR+1 (0).

Moreover, the C 0,1 norm of A˜ on RN depends on r, L, R and the C 0,1 norm of A on Ω. Finally, if A is uniformly elliptic with constant λ, also A˜ is uniformly elliptic with the same constant λ. Proof. We sketch the idea of the construction. We pick d˜0 and U as in Theorem 2.4 and we first extend A in Ω ∪ U as follows. We define, for any x ∈ Ω ∪ U ,  A(x) if x ∈ Ω ˜ A(x) = A(P∂Ω (x)) if x ∈ U \Ω. Then we fix a cutoff function χ ∈ C ∞ (R) such that χ is increasing, χ(t) = 0 for any t ≤ −3d˜0 /4 and χ(t) = 1 for any t ≥ 0. We extend A˜ all over RN as follows. We define, for any x ∈ RN , ˜ ˜ A(x) = χ(ϕ(x))A(x) + (1 − χ(ϕ(x)))IN . It is not difficult to check, with the help of Theorem 2.4, that such an extension satisfies the required properties. 

2.2

Riemannian manifolds

Let us consider the following definition of a Riemannian manifold M . Definition 2.7 Let Ω ⊂ RN be a bounded open set of class C 1,1 . Let G be a Lipschitz symmetric tensor in Ω which is uniformly elliptic with constant λ, 0 < λ < 1. For any x ∈ Ω, we denote as usual by gi,j (x) the elements of G(x) and by g i,j (x) the elements of G−1 (x), the inverse matrix of G(x). Finally, we set g(x) = | det(G(x))|. We call M the Riemannian manifold obtained by endowing Ω with the Lipschitz Riemannian metric whose tensor is given at any x ∈ Ω by gi,j (x)dxi ⊗ dxj . We finally say that G is a scalar metric if G = θIN with θ ∈ C 0,1 (Ω), that is, gi,j = θδi,j , where δi,j is the Kronecker delta. We recall the basic notation and properties of the Riemannian manifold M . At any point x ∈ Ω, given any two (column) vectors v and w, we denote hv, wiM = hG(x)v, wi and, consequently, kvkM = Clearly we have



p p hv, viM = hG(x)v, vi.

λkvk ≤ kvkM ≤



λ−1 kvk.

For any u ∈ L1 (Ω), we have Z Z p u(x) dM (x) = u(x) g(x) dx. Ω



9

If h ∈ L1 (∂Ω), with respect to the surface measure dσ, that is, with respect to the (N − 1)-dimensional Hausdorff measure, then p Z Z g(x) dσ(x), h(x) dσM (x) = h(x) α(x) ∂Ω ∂Ω where, for any x ∈ ∂Ω, 1 α(x) = p , −1 hG (x)ν(x), ν(x)i ν(x) being the outer normal to the boundary. We call νM (x) = α(x)G−1 (x)ν(x), which is the outer normal to the boundary with respect to the Riemannian metric. In fact, kνM (x)kM = 1 and hτ, νM (x)iM = 0 for any vector τ which is tangent to ∂Ω at the point x. At almost every x ∈ Ω, the intrinsic gradient of a function u ∈ W 1,1 (Ω) is defined by ∇M u(x) = ∇u(x)G−1 (x) = g i,j (x)

∂u (x)ej , ∂xi

where we used the summation convention. Let us note that, for any (column) vector v ∇u(x)v = h(∇u(x))T , vi = h(∇M u(x))T , viM . Therefore, (2.2) k∇M u(x)k2M = h(∇M u(x))T , (∇M u(x))T iM = h(∇u(x))T , (∇M u(x))T i = h(∇u(x))T , G−1 (x)(∇u(x))T i. Consequently, √ (2.3)

λk∇u(x)k ≤ k∇M u(x)kM ≤



λ−1 k∇u(x)k.

The intrinsic divergence of a vector field X ∈ W 1,1 (Ω, RN ) is defined, for almost every x ∈ Ω, by 1 √ divM X(x) = p div( gX)(x). g(x) For X ∈ W 1,1 (Ω, RN ), we have Z Z divM X(x) dM (x) = Ω

hX(x), νM (x)iM dσM (x).

∂Ω

Moreover, if X ∈ W 1,2 (Ω, RN ) and ψ ∈ W 1,2 (Ω), we have that 1 √ divM (Xψ) = √ div( gX)ψ + ∇ψX = divM (X)ψ + h(∇M ψ(x))T , XiM . g Finally, the following version of the coarea formula holds true. Let ϕ ∈ C 1 (Ω) be such that ∇ϕ 6= 0 everywhere. Then for any u ∈ L1 (Ω), we have ! Z Z Z u(x) u(x) dM (x) = dσM (x) dt. Ω R {x∈Ω: ϕ(x)=t} k∇M ϕ(x)kM 10

We call Γ = {γ : [0, 1] → Ω :R γ is piecewise C 1 }. For any curve γ ∈ Γ, we denote its 1 Euclidean length as length(γ) = 0 kγ 0 (t)k dt and, analogously, its Riemannian length as Z

1

lengthM (γ) =

kγ 0 (t)kM dt.

0

We have that

√ √ λ length(γ) ≤ lengthM (γ) ≤ λ−1 length(γ).

For any x and y ∈ Ω, we call Γ(x, y) = {γ ∈ Γ : γ(0) = x and γ(1) = y} and define d(x, y) =

inf

length(γ)

γ∈Γ(x,y)

Clearly



and dM (x, y) =

λ d(x, y) ≤ dM (x, y) ≤



inf γ∈Γ(x,y)

lengthM (γ).

λ−1 d(x, y),

whereas kx − yk ≤ d(x, y) ≤ C(Ω)kx − yk,

(2.4)

where C(Ω) is a constant depending on Ω only. If Ω satisfies the assumptions of Theorem 2.4, then C(Ω) depends on r, L and R only. We finally define the distance from the boundary in the Riemannian case. Let ϕM : Ω → R as follows. For any x ∈ Ω, ϕM (x) = distM (x, ∂Ω) = inf dM (x, y) = min dM (x, y). y∈∂Ω

y∈∂Ω

We observe that ϕ, the distance from the boundary in the Euclidean case that was defined in Definition 2.3, satisfies ϕ(x) = dist(x, ∂Ω) = inf d(x, y) = min d(x, y) y∈∂Ω

y∈∂Ω

for any x ∈ Ω

and, consequently, √

λ ϕ(x) ≤ ϕM (x) ≤



λ−1 ϕ(x)

for any x ∈ Ω.

As in the Euclidean case, we adopt the following notation. For any d ≥ 0, we define ΩdM = {x ∈ Ω : ϕM (x) > d}

and ∂ΩdM = {x ∈ Ω : ϕM (x) = d}.

Moreover, when d > 0, we call d UM = {x ∈ Ω : ϕM (x) < d}.

We recall that Theorem 2.4, which easily follows from [4, Theorem 5.7], contains the regularity properties of ϕ, the (signed) distance function from the boundary in the Euclidean case. For the Riemannian metric, a corresponding regularity result for ϕM is not easy to prove. We recall that fine regularity properties of the distance function from a general subset in a Riemannian manifold have been studied in [17]. In the next Section 3, we study the properties of the distance function from the boundary in the Riemannian case. 11

2.3

Definitions of frequencies of boundary data

Let Ω ⊂ RN be a bounded Lipschitz domain. By domain we mean, as usual, an open and connected set. We define the space of traces of H 1 (Ω) functions on ∂Ω as H 1/2 (∂Ω) = {f = u|∂Ω : u ∈ H 1 (Ω)}. We recall that H 1/2 (∂Ω) ⊂ L2 (∂Ω), with compact immersion. By Poincar´e inequality, an equivalent norm for H 1/2 (∂Ω), which we always adopt for simplicity, is given by the following kf k2H 1/2 (∂Ω) = kf k2L2 (∂Ω) + |f |2H 1/2 (∂Ω) ,

(2.5)

where the seminorm is given by |f |2H 1/2 (∂Ω)

(2.6)

Z =

k∇u0 (x)k2 dx



where u0 ∈ H 1 (Ω) is the weak solution to the following Dirichlet boundary value problem for the Laplace equation  ∆u0 = 0 in Ω (2.7) u0 = f on ∂Ω. Definition 2.8 We call frequency of a function f ∈ H 1/2 (∂Ω), with f 6= 0, the following quotient (2.8)

frequency(f ) =

|f |2H 1/2 (∂Ω) kf k2L2 (∂Ω)

for any f ∈ H 1/2 (∂Ω), f 6= 0.

R We denote L2∗ (∂Ω) = {ψ ∈ L2 (∂Ω) : ∂Ω ψ dσ = 0} and   Z 1/2 1/2 H∗ (∂Ω) = f ∈ H (∂Ω) : f dσ = 0 . ∂Ω

We call H −1/2 (∂Ω) the dual to H 1/2 (∂Ω) and −1/2

H∗

(∂Ω) = {η ∈ H −1/2 (∂Ω) : hη, 1i−1/2,1/2 = 0}.

By h·, ·i−1/2,1/2 we denote the duality between H −1/2 (∂Ω) and H 1/2 (∂Ω). By Poincar´e inequality, we have that kf kH 1/2 (∂Ω) = |f |H 1/2 (∂Ω) ∗

1/2

for any f ∈ H∗ (∂Ω)

1/2

is an equivalent norm for H∗ (∂Ω) and, analogously, kηkH −1/2 (∂Ω) = ∗

sup kψk

=1 1/2 H∗ (∂Ω)

−1/2

is an equivalent norm for H∗

hη, ψi−1/2,1/2

(∂Ω). 12

−1/2

for any η ∈ H∗

(∂Ω)

We observe that any η ∈ L2 (∂Ω) is considered as an element of H −1/2 (∂Ω) by setting Z ηψ dσ for any ψ ∈ H 1/2 (∂Ω). (2.9) hη, ψi−1/2,1/2 = ∂Ω −1/2

Moreover, if η ∈ L2∗ (∂Ω) then η ∈ H∗ (∂Ω). It is important to note that here, and in the definitions of L2 (∂Ω) and L2∗ (∂Ω), we use the usual (N − 1)-dimensional Hausdorff measure on ∂Ω. In the sequel we adopt the same convention even if Ω is endowed with a Riemannian metric G which is different from the Euclidean one. This simplifies the treatment of certain changes of variables for the Neumann problem or for the Steklov eigenvalue problem, see Remark 2.15. 1/2

Definition 2.9 We call lower frequency of a function f ∈ H∗ (∂Ω), with f 6= 0, the following quotient (2.10)

kf k2L2 (∂Ω)

lowfrequency(f ) =

kf k2

−1/2

H∗

Here

1/2

for any f ∈ H∗ (∂Ω), f 6= 0.

(∂Ω)

Z

Z kf kH −1/2 (∂Ω) = ∗

f ψ dσ =

sup kψk

=1 1/2 H∗ (∂Ω)

f ψ dσ.

sup |ψ|H 1/2 (∂Ω) =1 ∂Ω

∂Ω

1/2

From this definition, we immediately infer that, for any f ∈ H∗ (∂Ω), with f 6= 0, we have kf k4L2 (∂Ω) ≤ kf k2 −1/2 kf k2 1/2 = kf k2 −1/2 |f |2H 1/2 (∂Ω) H∗

(∂Ω)

H∗

(∂Ω)

H∗

(∂Ω)

hence (2.11)

2.4

lowfrequency(f ) ≤ frequency(f )

1/2

for any f ∈ H∗ (∂Ω), f 6= 0.

Boundary value problems for elliptic equations

Let Ω ⊂ RN be a bounded Lipschitz domain. We consider Dirichlet and Neumann problems in Ω for elliptic equations in divergence form, in the Euclidean and in the Riemannian setting. Let A = A(x) be a conductivity tensor in Ω, that is, A is a symmetric tensor in Ω which is uniformly elliptic with some constant λ1 , 0 < λ1 < 1. If A = γIN , where γ ∈ L∞ (Ω) satisfies λ1 ≤ γ(x) ≤ λ−1 for a.e. x ∈ Ω, 1 we say that A (or γ) is a scalar conductivity. We say that a conductivity tensor A is Lipschitz if A is a Lipschitz symmetric tensor. Analogously, A (or γ) is a Lipschitz scalar conductivity if γ ∈ C 0,1 (Ω). Let G be a Lipschitz symmetric tensor in Ω which is uniformly elliptic with constant λ, 0 < λ < 1, and let M be the corresponding Riemannian manifold on Ω as in Definition 2.7. In this subsection we adopt the following assumption. Assumption 2.10 We assume that either A is a scalar conductivity tensor, that is, A = γIN with γ ∈ L∞ (Ω), or G is a scalar metric, that is, G = θIN with θ ∈ C 0,1 (Ω). 13

For any f ∈ H 1/2 (∂Ω), let u ∈ H 1 (Ω) be the weak solution to the Dirichlet boundary value problem  divM (A∇M u) = 0 in Ω (2.12) u=f on ∂Ω. We recall that u ∈ H 1 (Ω) solves (2.12) if u = f on ∂Ω in the trace sense and Z hA(x)(∇M u(x))T , (∇M ψ(x))T iM dM (x) = 0 for any ψ ∈ H01 (Ω). Ω

For the sake of simplicity, we sometimes drop the transpose in the sequel, considering, with a small abuse of notation, the gradient as a column vector. The following remark holds true. Remark 2.11 Let u and u0 be the solution to (2.12) and (2.7), respectively. Then there exists a constant c1 , 0 < c1 < 1 depending on λ and λ1 only, such that Z Z Z −1 2 (2.13) c1 k∇u0 (x)k dx ≤ hA(x)∇M u(x), ∇M u(x)iM dM (x) ≤ c1 k∇u0 (x)k2 dx. Ω





In fact, on the one hand, by the Dirichlet principle, Z Z Z 2 hA(x)∇M u(x), ∇M u(x)iM dM (x) ≥ c1 k∇u(x)k dx ≥ c1 k∇u0 (x)k2 dx. Ω





On the other hand, correspondingly we have Z

Z hA(x)∇M u(x), ∇M u(x)iM dM (x) ≤





hA(x)∇M u0 (x), ∇M u0 (x)iM dM (x) Z k∇u0 (x)k2 dx. ≤ c−1 1 Ω

As a consequence of Remark 2.11, we can define equivalent H 1/2 (∂Ω) norm and seminorm which are given by, for any f ∈ H 1/2 (∂Ω), Z 2 (2.14) |f | 1/2 = hA(x)∇M u(x), ∇M u(x)iM dM (x), HA (∂Ω)



where u solves (2.12), and (2.15)

kf k2

1/2

HA (∂Ω)

= kf k2L2 (∂Ω) + |f |2

1/2

HA (∂Ω)

.

We can also define an equivalent H −1/2 (∂Ω) norm given by, for any η ∈ H −1/2 (∂Ω), kηkH −1/2 (∂Ω) = A

sup

hη, ψi−1/2,1/2 .

kψk

=1 1/2 H (∂Ω) A

We note that here we drop the dependence on the metric M , although the seminorm, and thus the norms as well, clearly also depends on it. 14

Analogously, kf kH 1/2 (∂Ω) = |f |H 1/2 (∂Ω) ∗,A

1/2

for any f ∈ H∗ (∂Ω)

A

1/2

is an equivalent norm for H∗ (∂Ω) and kηkH −1/2 (∂Ω) = ∗,A

hη, ψi−1/2,1/2

sup

−1/2

for any η ∈ H∗

(∂Ω)

kψk

=1 1/2 H (∂Ω) ∗,A

−1/2

is an equivalent norm for H∗ (∂Ω). −1/2 For any η ∈ H∗ (∂Ω), let v ∈ H 1 (Ω) be the solution to the Neumann boundary value problem   divM (A∇M v) = 0 in Ω hA∇M v, νM iM = η on ∂Ω (2.16)  R ∂Ω v dσ = 0. 1/2

By a solution we mean v ∈ H 1 (Ω) such that v|∂Ω ∈ H∗ (∂Ω) and that Z hA∇M v, ∇M ψiM = hη, ψ|∂Ω i−1/2,1/2 for any ψ ∈ H 1 (Ω). Ω

We also note that, for simplicity and by a slight abuse of notation, we denote AuνM = hA∇M v, νM iM . Such a notation is actually correct when A = γIN is a scalar conductivity. In fact, in this case, AuνM = γuνM where uνM is the (exterior) normal derivative of u with respect to Ω which, in the Riemannian setting, is given by uνM = h(∇M u)T , νM iM = ∇uνM . By Poincar´e inequality and Lax-Milgram lemma, we have that there exists a unique solution both to (2.12) and to (2.16). Moreover, there exists a constant c2 , 0 < c2 < 1 depending on Ω, λ and λ1 only, such that for any f ∈ H 1/2 (∂Ω) c2 kf kH 1/2 (∂Ω) ≤ kukH 1 (Ω) ≤ c−1 2 kf kH 1/2 (∂Ω) −1/2

and for any η ∈ H∗

(∂Ω)

c2 kηkH −1/2 (∂Ω) ≤ kvkH 1 (Ω) ≤ c−1 2 kηkH −1/2 (∂Ω) . If Ω ⊂ BR (0) is Lipschitz with positive constants r and L, the dependence of c2 on Ω is just through the constants r, L and R. −1/2 Let Λ : H 1/2 (∂Ω) → H∗ (∂Ω) be the linear operator such that Λ(f ) = hA∇M v, νM iM

for any f ∈ H 1/2 (∂Ω)

where u solves (2.12). Here, we mean Z ˜ −1/2,1/2 = hA∇M u, ∇M ψiM dM hhA∇M v, νM iM , ψi Ω

15

for any ψ˜ ∈ H 1/2 (∂Ω),

1/2

˜ We infer that Λ restricted to H∗ (∂Ω) where ψ is any H 1 (Ω) function such that ψ|∂Ω = ψ. −1/2 1/2 is invertible and both Λ and Λ−1 : H∗ (∂Ω) → H∗ (∂Ω) are bounded operators with norms bounded by constants depending on Ω, λ and λ1 only. As usual we refer to Λ as the Dirichlet-to-Neumann map and to Λ−1 as the Neumann-to-Dirichlet map. We are interested in eigenvalues and eigenfunctions of the Dirichlet-to-Neumann map Λ, which coincides with the so-called Steklov eigenvalues and eigenfunctions. Namely, we say that µ ∈ C and φ ∈ L2 (∂Ω), with φ 6= 0 are, respectively, a Steklov eigenvalue and its corresponding eigenfunction if there exists w ∈ H 1 (Ω) such that w = φ on ∂Ω and w satisfies  divM (A∇M w) = 0 in Ω (2.17) hA∇M v, νM iM = µw on ∂Ω, that is, Z Z hA∇M w, ∇M ψiM dM = hµw|∂Ω , ψ|∂Ω i−1/2,1/2 = Ω

for any ψ ∈ H 1 (Ω).

µwψ dσ

∂Ω

In other words, φ satisfies Λ(φ) = µφ. Clearly (2.17) is satisfied by µ = 0 and w a constant function. It is well-known that the Steklov eigenvalues form an increasing sequence of real numbers 0 = µ0 < µ1 ≤ µ2 ≤ . . . ≤ µn ≤ . . . such that limn µn = +∞. For any n ≥ 0, we can find a corresponding eigenfunction φn , normalised in such a way that kφn kL2 (∂Ω) = 1, such that {φn }n≥0 is an orthonormal basis of √ L2 (∂Ω) and {φn }n∈N is an orthonormal basis of L2∗ (∂Ω). Moreover, {φn / 1 + µn }n≥0 and √ 1/2 {φn / 1 + µn }n∈N are an orthonormal basis of H 1/2 (∂Ω) and H∗ (∂Ω), respectively, with √ 1/2 respect to the HA (∂Ω) norm. Finally, we call {ψn = φn / µn }n∈N and we note that it is 1/2

1/2

an orthonormal basis of H∗ (∂Ω) with respect to the H∗,A (∂Ω) norm. 1/2

If φ ∈ H∗ (∂Ω) is a Steklov eigenfunction with eigenvalue µ, and w is the corresponding solution to (2.17), then R R µφ2 dσ hA∇M w, ∇M wiM dM ∂Ω R µ= R = Ω , 2 2 ∂Ω φ dσ ∂Ω φ dσ hence by Remark 2.11 we have, with the same constant c1 , c1 frequency(φ) ≤ µ ≤ c−1 1 frequency(φ).

(2.18)

An important property of Steklov eigenfunctions is that their frequency and lower frequency R 1/2 are of the same order. In fact, for µ > 0 we have φ ∈ H∗ (∂Ω) and, setting ∂Ω φ2 = 1, c1 |φ|2H 1/2 (∂Ω) ≤ kφk2

1/2

H∗,A (∂Ω)

2 = µ ≤ c−1 1 |φ|H 1/2 (∂Ω) ,

therefore c1 kφk2

−1/2

H∗

(∂Ω)

≤ kφk2

−1/2

H∗,A (∂Ω)

2 = µ−1 ≤ c−1 1 kφk

−1/2

H∗

(∂Ω)

,

and, finally, (2.19)

c1 lowfrequency(φ) ≤ µ ≤ c−1 1 lowfrequency(φ).

Although their proofs are elementary, and actually quite similar, the next two remarks are crucial. 16

Remark 2.12 Let A be a conductivity tensor in Ω which is uniformly elliptic with some constant λ1 , 0 < λ1 < 1. Let G be a Lipschitz symmetric tensor in Ω which is uniformly elliptic with constant λ, 0 < λ < 1, and let M be the corresponding Riemannian manifold on Ω. Let Assumption 2.10 be satisfied. Let us take η1 ∈ C 0,1 (Ω) such that λ1 ≤ η1 ≤ λ−1 1 in Ω, for some constant λ1 , 0 < λ1 < 1. ˜ ˜ obtained by endowing Let us define G = η1 G and let us consider the Riemannian manifold M ˜ Ω with the Lipschitz Riemannian metric given by G. We define (2−N )/2 A˜ = η1 A, and we note that A˜ = A if N = 2. Then, for any ψ1 , ψ2 ∈ H 1 (Ω) we have Z Z ˜ ˜ ψ1 , ∇ ˜ ψ2 i ˜ d ˜ . hA∇M ψ1 , ∇M ψ2 iM dM = hA∇ M M M M Ω



The next remark shows that, under Assumption 2.10 and if A is Lipschitz, we can always assume that the conductivity tensor is a scalar conductivity, up to changing the Riemannian metric. For example, this applies when A is a Lipschitz conductivity tensor and the metric is the Euclidean one. Namely we have the following. Remark 2.13 Let A be a Lipschitz conductivity tensor in Ω which is uniformly elliptic with some constant λ1 , 0 < λ1 < 1. Let G be a Lipschitz symmetric tensor in Ω which is uniformly elliptic with constant λ, 0 < λ < 1, and let M be the corresponding Riemannian manifold on Ω. Let Assumption 2.10 be satisfied. √ We call A1 = gAG−1 and γ1 = (det A1 )1/N so that A1 = γ1 Aˆ1 with det Aˆ1 ≡ 1. If N > 2, we define A˜ ≡ IN and ˜ = (det(A1 ))1/(N −2) A−1 . G 1 If N = 2, we define A˜ ≡ γ1 IN and ˜ = Aˆ−1 . G 1 ˜ obtained by endowing Ω with the Lipschitz Let us consider the Riemannian manifold M ˜ Riemannian metric given by G. Then, for any ψ1 , ψ2 ∈ H 1 (Ω) we have Z Z ˜ ˜ ψ1 , ∇ ˜ ψ2 i ˜ d ˜ . hA∇M ψ1 , ∇M ψ2 iM dM = hA∇ M M M M Ω



Both for the case of Remark 2.12 and the one of Remark 2.13, we infer the following consequences. Fixed f ∈ H 1/2 (∂Ω), let u be the solution to (2.12). Then u solves  ˜ ˜ u) = 0 in Ω divM˜ (A∇ M (2.20) u=f on ∂Ω. −1/2

Analogously, fixed η ∈ H∗ (2.21)

(∂Ω), let v be the solution to (2.16). Then v solves  ˜ ˜ v) = 0 in Ω  divM˜ (A∇ M ˜ ˜ v, ν ˜ i ˜ = η on ∂Ω hRA∇ M M M  v dσ = 0. ∂Ω 17

Finally, if w solves (2.17) for a constant µ, then w solves  ˜ ˜ w) = 0 divM˜ (A∇ in Ω M (2.22) ˜ hA∇M˜ w, νM˜ iM˜ = µw on ∂Ω. We conclude this section by investigating the regularity of the solutions to (2.12), (2.16) and (2.17). We need stronger assumptions on the domain Ω and the conductivity tensor A. Namely we assume the following till the end of the section. Let us fix positive constants R, r, L, C0 , C1 , λ and λ1 , with 0 < λ < 1 and 0 < λ1 < 1. We refer to these constants as the a priori data. Let Ω ⊂ BR (0) ⊂ RN be a bounded domain of class C 1,1 with constants r and L. Let G be a Lipschitz symmetric tensor in Ω which is uniformly elliptic with constant λ and such that kGkC 0,1 (Ω) ≤ C0 . Let A be a Lipschtitz conductivity tensor in Ω which is uniformly elliptic with constant λ1 and such that kAkC 0,1 (Ω) ≤ C1 . We suppose that Assumption 2.10 holds. We note that, without loss of generality, through Remark 2.13, we could just assume that A is a scalar conductivity. The first remark is that, by standard regularity estimates for elliptic equations, if u is 2 (Ω) and the equation is satisfied any weak solution to divM (A∇M u) = 0 in Ω, then u ∈ Hloc pointwise almost everywhere in Ω. Here we are interested on the conditions that guarantee that our solutions are actually belonging to H 2 (Ω). 3/2 We adopt the standard definition of H 3/2 (∂Ω), see for example [13], and by H∗ (∂Ω) we denote the elements of H 3/2 (∂Ω) with zero mean on ∂Ω. Let u be the solution to (2.12) with boundary datum f ∈ H 1/2 (∂Ω) and v the solution to (2.16) with boundary datum −1/2 η ∈ H∗ (∂Ω). The following regularity properties hold true. Proposition 2.14 There exist a positive constants c3 , 0 < c3 < 1 depending on the a priori data only, such that for any f ∈ H 3/2 (∂Ω) c3 kf kH 3/2 (∂Ω) ≤ kukH 2 (Ω) ≤ c−1 3 kf kH 3/2 (∂Ω)

(2.23) 1/2

and for any η ∈ H∗ (∂Ω) c3 kηkH 1/2 (∂Ω) ≤ kvkH 2 (Ω) ≤ c−1 3 kηkH 1/2 (∂Ω) .

(2.24)

In (2.24), we can replace kηkH 1/2 (∂Ω) with kηkH 1/2 (∂Ω) , kηkH 1/2 (∂Ω) or kηkH 1/2 (∂Ω) . ∗

3/2 H∗ (∂Ω)

∗,A

A

1/2 H∗ (∂Ω),

As a consequence, Λ is bounded between and with a bounded inverse, and their norms are bounded by constants depending on the a priori data only. Before sketching the proof of this standard regularity result, we state the following important remark. Remark 2.15 Let v ∈ H 2 (Ω) be a solution to divM (A∇M v) = 0 in Ω. Then ∇v ∈ H 1 (Ω), therefore ∇v is well-defined, in the trace sense, on ∂Ω. It follows that AvνM is well-defined for instance in L2 (∂Ω). Moreover, using integration by parts, we conclude that for any ψ ∈ H 1 (Ω) we have Z Z Z hA∇M v, ∇M ψiM dM = AvνM ψ dσM = ηψ dσ Ω

∂Ω

18

∂Ω

where (2.25)

η=

√ g √ AvνM = ghA∇M v, νi. α

Therefore, in the Riemannian setting, the Neumann condition AvνM = η

on ∂Ω

is in general not valid in a pointwise or L2 sense, even when both AvνM and η are welldefined as L2 (∂Ω) functions. The correct pointwise or L2 boundary condition is given in (2.25). Proof of Proposition 2.14. This result is essentially proved in [13]. Using for instance [13, Theorem 1.5.1.2] and [13, Theorem 1.5.1.3], with the help of Remark 2.15, we immediately infer that the left inequalities of (2.23) and (2.24) hold true. The right inequalities of (2.23) and (2.24) easily follow by [13, Corollary 2.2.2.4] and [13, Corollary 2.2.2.6].  An important consequence of Proposition 2.14 for Steklov eigenfunctions is the following. 1/2

Corollary 2.16 Let φ ∈ H∗ (∂Ω) be a Steklov eigenfunction with eigenvalue µ > 0 and let w be the corresponding solution to (2.17). Then (2.26)

−1 2 2 c23 µ2 (1 + c1 µ)kφk2L2 (∂Ω) ≤ kwk2H 2 (Ω) ≤ c−2 3 µ (1 + c1 µ)kφkL2 (∂Ω) ,

where c1 is as in (2.18) and c3 is as in Proposition 2.14, thus they depend on the a priori data only. Proof. By (2.18), we have that 2 c1 µkφk2L2 (∂Ω) ≤ |φ|2H 1/2 (∂Ω) ≤ c−1 1 µkφkL2 (∂Ω) .

Therefore 2 (1 + c1 µ)kφk2L2 (∂Ω) ≤ kφk2H 1/2 (∂Ω) ≤ (1 + c−1 1 µ)kφkL2 (∂Ω) .

Then the result follows by Proposition 2.14, in particular by (2.24) with η = µφ. Finally, we state and prove the following result.



Proposition 2.17 There exists a constant C2 , depending on the a priori data only, such that for any f ∈ H 1/2 (∂Ω) we have kukL2 (Ω) ≤ C2 kf kH −1/2 (∂Ω) ,

(2.27)

where u is the solution to (2.12). 1/2

Proof. Without loss of generality, we can restrict our attention to f ∈ H∗ (∂Ω) and we −1/2 1/2 can replace the H −1/2 (∂Ω) norm with the H∗,A (∂Ω) norm. Given f ∈ H∗ (∂Ω), we can find a sequence {αn }n∈N of real numbers such that X X f= αn ψn and kf k2 1/2 = αn2 . H∗,A (∂Ω)

n∈N

19

n∈N

Furthermore, it is easy to infer that kf k2

=

−1/2 H∗,A (∂Ω)

X α2 n . µ2n

n∈N

We have that X

Λ(f ) =

αn µn ψn ,

n∈N

therefore kΛ(f )k2

=

1/2 H∗,A (∂Ω)

X

αn2 µ2n .

n∈N 1/2

By Proposition 2.14, in particular by (2.24), for any f ∈ H∗ (∂Ω), we have that X X c23 αn2 µ2n ≤ kuk2H 2 (Ω) ≤ c−2 αn2 µ2n , 3 n∈N

n∈N

possibly for a different constant 0 < c3 < 1 still depending on the a priori data only. 1/2 2 Let P us now consider a function v ∈ H (Ω) such that h = v|∂Ω ∈ H∗ (∂Ω). In particular h = n∈N βn ψn for a suitable sequence {βn }n∈N of real numbers. We call v˜ the solution to (2.12) with boundary datum given by h. Then Z Z X hA∇M u, ∇M viM dM = hA∇M u, ∇M v˜iM dM = αn βn . Ω



n∈N

If we call, for any n ∈ N, β˜n = βn µn , then sup P ˜2 n βn ≤1

X n∈N

β˜n αn µn

!

X αn β˜n µn

= P sup

˜2 n βn ≤1

!

n∈N

=

X α2 n µ2n

!1/2 .

n∈N

1/2

In other words, for any v ∈ H 2 (Ω) with h = v|∂Ω ∈ H∗ (∂Ω) we have !1/2

Z hA∇M u, ∇M viM dM ≤ kf k −1/2 H∗,A (∂Ω)

X



βn2 µ2n

n∈N

≤ c−1 v kH 2 (Ω) 3 kf kH −1/2 (∂Ω) k˜ ∗,A

−1 ≤ c−2 3 kf kH −1/2 (∂Ω) khkH 3/2 (∂Ω) ≤ c4 kf kH −1/2 (∂Ω) kvkH 2 (Ω) , ∗,A

∗,A

where 0 < c4 < 1 is a constant still depending on the a priori data only. Now, for any ϕ ∈ L2 (Ω), let w be the weak solution to   divM (A∇M w) = ϕ in Ω hA∇M w, νM iM = c on ∂Ω (2.28)  R ∂Ω w dσ = 0, where the constant c is such that Z

Z c dσ =

∂Ω

ϕ(x) dx. Ω

20

1/2

By a solution we mean w ∈ H 1 (Ω) such that w|∂Ω ∈ H∗ (∂Ω) and that Z Z Z ϕ(x)ψ(x) dx for any ψ ∈ H 1 (Ω). cψ dσ − hA∇M w, ∇M ψiM = Ω

∂Ω



Still by standard regularity estimates, see for instance [13, Chapter 2], we have that kwkH 2 (Ω) ≤ C3 kϕkL2 (Ω) , where C3 is a constant depending on the a priori data only. We conclude that, for any ϕ ∈ L2 (Ω), Z Z u(x)ϕ(x) dx = hA∇M u, ∇M wiM dM Ω



−1 ≤ c−1 4 kf kH −1/2 (∂Ω) kwkH 2 (Ω) ≤ C3 c4 kf kH −1/2 (∂Ω) kϕkL2 (Ω) , ∗,A

∗,A

therefore kukL2 (Ω) ≤ C3 c−1 4 kf kH −1/2 (∂Ω) ∗,A

and the proof is concluded.

3



The distance function from the boundary

Let M be a Riemannian manifold as in Definition 2.7. We begin by investigating the consequences of assuming that ϕM is smooth enough, namely we consider the following. Assumption 3.1 For M , a Riemannian manifold as in Definition 2.7, we assume that there d0 exists d0 > 0 such that ϕM ∈ C 1,1 (UM ). The first consequence of Assumption 3.1 is the following. Proposition 3.2 Under Assumption 3.1, we have d0 k∇M ϕkM = 1 in UM .

(3.1)

Proof. We divide the proof into several steps. First step. We show that ∇ϕM is different from 0 on ∂Ω. In fact, for any x ∈ ∂Ω we have −

∂ϕM ϕM (x − tν(x)) − ϕM (x) ϕM (x − tν(x)) (x) = lim = lim ∂ν(x) t t t→0+ t→0+ √ √ ϕ(x − tν(x)) ϕ(x − tν(x)) − ϕ(x) √ ≥ λ lim = λ lim = λ > 0. t t t→0+ t→0+

In the last equality we used (2.1). d0 Second step. We prove that k∇M ϕM kM ≤ 1 in UM . This follows from the obvious fact that ϕM is Lipschitz with Lipschitz constant 1 with respect to the distance dM , that is,

|ϕM (x) − ϕM (y)| ≤ dM (x, y) 21

for any x, y ∈ Ω.

d0 Then, let x ∈ UM and let γ : [0, 1] → Ω be a C 1 curve such that γ(0) = x and γ 0 (0) = v, with v = −ν(x) if x ∈ ∂Ω. We have

d (ϕM ◦ γ)(0) = ∇ϕM (x)v = h(∇M ϕM (x))T , viM . dt On the other hand, d (ϕM ◦ γ)(0) = lim |ϕM (γ(t)) − ϕM (γ(0))| dt t→0+ t dM (γ(t), γ(0)) ≤ lim ≤ lim t t→0+ t→0+

Rt 0

kγ 0 (s)kM ds = kγ 0 (0)kM = kvkM . t

Thus, for any v or for v = −ν(x) if x ∈ ∂Ω, we have h(∇M ϕM (x))T , viM ≤ kvkM , hence k∇M ϕM (x)kM ≤ 1. Third step. By the first step and continuity, there exists d1 , 0 < d1 ≤ d0 , such that we have d1 0 < k∇M ϕM (x)kM ≤ 1 for any x ∈ UM . d1 We show that k∇M ϕM (x)kM = 1 for any x ∈ UM . By contradiction, we assume there d1 d1 exist x0 ∈ UM , r > 0 and 0 < c < 1 such that Br (x0 ) ⊂ UM and 0 < k∇M ϕM (y)kM ≤ c for any y ∈ Br (x0 ). In particular, there exists 0 < t0 such that y0 = x0 + t0 ∇ϕM ∈ Br (x0 ) and it satisfies the following conditions ϕM (y0 ) > ϕM (x0 )

and

d1 2dM (x0 , y0 ) ≤ dM (y0 , y) for any y ∈ UM \Br (x0 ).

We call h = ϕM (y0 ) − ϕM (x0 ) and we obviously have 0 < h ≤ dM (x0 , y0 ). Finally, we fix ε such that   1−c 0 < ε < min 1, h and ϕM (y0 ) + ε < d1 . c d1 Let γ ∈ Γ be such that γ([0, 1]) ⊂ UM , γ(0) = y0 , γ(1) ∈ ∂Ω and

lengthM (γ) ≤ ϕM (y0 ) + ε < d1 . There must be s0 , 0 < s0 ≤ 1, such that ϕM (γ(s0 )) = ϕM (x0 ). Therefore h = ϕM (γ(0)) − ϕM (γ(s0 )) ≤ dM (γ(0), γ(s0 )) ≤ lengthM (γ([0, s0 ])) ≤ h + ε. But γ([0, s0 ])) ⊂ Br (x0 ), otherwise 0 < 2h ≤ 2dM (x0 , y0 ) ≤ lengthM (γ([0, s0 ])) ≤ h + ε < 2h which leads to a contradiction. Therefore, Z s0 0 h = ϕM (γ(0)) − ϕM (γ(s0 )) = − ∇ϕM (γ(t))γ (t) dt 0 Z s0 Z T 0 = h(∇M ϕM (γ(t))) , γ (t)iM dt ≤ 0

s0

ckγ 0 (t)kM dt

0

= c lengthM (γ([0, s0 ])) ≤ c(h + ε) < h 22

d1 which leads to a contradiction, thus k∇M ϕM (x)kM = 1 for any x ∈ UM .

Fourth step. Let d d2 = sup{d : 0 < d ≤ d0 and k∇M ϕM (x)kM = 1 for any x ∈ UM }.

By the third step we have d1 ≤ d2 . If d2 = d0 then the result is proved. Assume, by contradiction, that d2 < d0 . Then, by continuity, there exists d, d2 < d < d0 , such that d . By the same reasoning used in the third step, we 0 < k∇M ϕM (x)kM ≤ 1 for any x ∈ UM d conclude that k∇M ϕM kM = 1 in UM , which contradicts the definition of d2 .  Under Assumption 3.1, we have that, for any 0 ≤ d < d0 , ΩdM is a C 1,1 open set and ∂(ΩdM ) = ∂ΩdM . Let ν denote the exterior normal to ΩdM on ∂ΩdM , and νM its corresponding one in the Riemannian setting. Then we have. Proposition 3.3 Under Assumption 3.1, for any 0 ≤ d < d0 , we have (∇M ϕM )T = −νM on ∂ΩdM .

(3.2)

In particular this is true on ∂Ω. Proof. It is clear that, for any x ∈ ∂ΩdM , we have (∇ϕM (x))T = −a(x)ν(x) for some positive constant a(x) depending on x. By the definitions of ∇M ϕM (x) and of νM (x), we easily conclude that (∇M ϕM (x))T = −a1 (x)νM (x) for some positive constant a1 (x) depending on x. Since, by Proposition 3.2, k∇M ϕM (x)kM = kνM (x)kM = 1, the result immediately follows.  d0 Remark 3.4 Under Assumption 3.1, if kϕM kC 1,1 (U d0 ) ≤ C0 , then UM is a C 1,1 open set M with constants r1 and L1 depending on r, L, R, d0 and C0 only. This result can be obtained d0 by an approximation argument, namely by suitably approximating ∂UM ∩ Ω with ∂ΩdM as d → d− 0.

The key point is the following complementary result. d0 Proposition 3.5 Fixed d0 > 0, let f ∈ C 1,1 (UM ) be a nonnegative function such that d0 k∇M f kM = 1 in UM

and

f = 0 on ∂Ω.

d0 Then f = ϕM on UM . Moreover, if kf kC 1,1 (U d0 ) ≤ C0 , we have that M

(3.3)

k∇M f kC 0,1 (U d0 ,RN ) ≤ C1 , M

with C1 depending on C0 , λ and the Lipschitz constant of the metric G only. d0 , for any x ∈ ∂Ω Proof. First of all, we note that, since f = 0 on ∂Ω and f ≥ 0 in UM T we have (∇f (x)) = −a(x)ν(x) for some positive constant a(x) depending on x, thus, reasoning as in Proposition 3.3, (∇M f )T = −νM on ∂Ω. We can also easily conclude that d0 f > 0 on UM \∂Ω.

23

d0 Let x ∈ UM \∂Ω. Fixed y ∈ ∂Ω, let γ ∈ Γ(y, x). Without loss of generality we can assume d0 that γ([0, 1]) ⊂ UM . Then 1

Z

∇f (γ(s))γ 0 (s) ds

f (x) = f (x) − f (y) = 0

1

Z

T

0

Z

1

h(∇M f (γ(s))) , γ (s)iM ds ≤

= 0

kγ 0 (s)kM ds = lengthM (γ).

0

We can conclude that f (x) ≤ ϕM (x)

(3.4)

d0 for any x ∈ UM .

Since ∇f is Lipschitz, by the definition of ∇M f and the properties of G, we immediately infer that also ∇M f is Lipschitz. Analogously, one can prove (3.3). d0 For any x ∈ UM , let γx be the (maximal) solution to the Cauchy problem for the ordinary differential equation  0 γx = (∇M f (γx ))T , γx (0) = x. Since ∇M f is Lipschitz, we have existence and uniqueness of a solution γx for t ∈ [0, T ), d0 for some suitable T > 0 depending on x, even if x ∈ ∂Ω. Moreover, for any x ∈ UM , d0 γx (t) ∈ UM \∂Ω for any 0 < t < T . d0 Let x ∈ UM . For any t0 , t1 ∈ R such that t0 < t1 and for which γx is defined, let us call z0 = γx (t0 ) and z1 = γx (t1 ). Then we observe that Z

t1

f (z1 ) − f (z0 ) =

∇f (γx (s))γx0 (s) ds

t0

Z

t1

=

h(∇M f (γx (s)))T , γx0 (s)iM ds = lengthM (γx ([t0 , t1 ])) = t1 − t0 ,

t0

therefore dM (z0 , z1 ) ≤ f (z1 ) − f (z0 ). In particular, if z0 ∈ ∂Ω, then ϕM (z1 ) ≤ dM (z0 , z1 ) ≤ f (z1 ) − f (z0 ) = f (z1 ), thus, by the previous inequality (3.4), we have ϕM (z1 ) = f (z1 ). We claim the following result. Let d1 , 0 ≤ d1 < d0 be such that f (x) = ϕM (x) for any d0 x ∈ UM with ϕM (x) ≤ d1 . Then there exists d such that d1 < d < d0 and f (x) = ϕM (x) d0 for any x ∈ UM with ϕM (x) ≤ d. In order to prove the claim, let us begin with the following remark, where we assume d0 that d1 > 0. Let x ∈ UM be such that ϕM (x) = f (x) = d1 . By the implicit function theorem, there exist a C 1 function φx : RN −1 → R and an open neighbourhood Ux of x such that for any y ∈ Ux we have, up to a rigid transformation depending on x, n o n o (3.5) y ∈ Ux : f (y) S d1 = y = (y 0 , yN ) ∈ Ux : yN S φx (y 0 ) . Without loss of generality, up to changing Ux , we can assume that  Ux− = y = (y 0 , yN ) ∈ Ux : yN < φx (y 0 ) 24

is connected. We want to show that (3.5) holds true even if we replace f with ϕM . By (3.4), it is clear that  {y ∈ Ux : ϕM (y) > d1 } ⊃ {y ∈ Ux : f (y) > d1 } = y = (y 0 , yN ) ∈ Ux : yN > φx (y 0 ) . Moreover, by our assumption,  {y ∈ Ux : ϕM (y) 5 d1 } ⊂ {y ∈ Ux : f (y) 5 d1 } = y = (y 0 , yN ) ∈ Ux : yN 5 φx (y 0 ) . Since ϕM can not have interior local minimum points, there exists y1 ∈ Ux such that ϕM (y1 ) < d1 . Then f (y1 ) < d1 and y1 ∈ Ux− . Assume by contradiction that there exists y2 ∈ Ux such that ϕM (y2 ) > d1 ≥ f (y2 ). Actually, by continuity, we can always assume that ϕM (y2 ) > d1 > f (y2 ), hence that y2 ∈ Ux− as well. We connect y1 to y2 with a smooth curve all contained in Ux− . There must be a point y along this curve on which ϕM (y) = d1 , thus we obtain a contradiction since f (y) < d1 . This remark allows us to show that there exists ε > 0 such that d1 + ε < d0 and f (y) > d1 for any y with d1 < ϕM (y) < d1 + ε. Assume by contradiction that there exists y0 such that d1 < f (y0 ) < ϕM (y0 ) ≤ d1 + ε/4. We note that it is well-defined z0 = γy0 (d1 − f (y0 )). It happens that f (z0 ) = d1 and dM (z0 , y0 ) ≤ (f (y0 ) − d1 ) ≤ ε/4, therefore ϕM (z0 ) ≤ d1 + ε/2. By (3.4), d1 ≤ ϕM (z0 ) but ϕM (z0 ) can not be greater than d1 , otherwise f (z0 ) should be greater than d1 as well. We conclude that ϕM (z0 ) = d1 , therefore ϕM (y0 ) ≤ ϕM (z0 ) + dM (z0 , y0 ) ≤ d1 + (f (y0 ) − d1 ) = f (y0 ) which gives the contradiction and proves the claim. Let us conclude the proof by defining d0 d2 = sup{d : 0 < d < d0 and f (x) = ϕM (x) for any x ∈ UM with ϕM (x) ≤ d}.

If d2 = d0 the proof is concluded. If, by contradiction, d2 < d0 , by continuity we have that d0 f (x) = ϕM (x) for any x ∈ UM with ϕM (x) ≤ d2 and the claim contradicts the definition of d2 .  We point out the following important property. Under Assumption 3.1, or equivalently d0 under the assumptions of Proposition 3.5, for any x ∈ UM , let γx be the (maximal) solution to the Cauchy problem for the ordinary differential equation  0 γx = (∇M ϕM (γx ))T , (3.6) γx (0) = x. Then γx : [−ϕM (x), d0 − ϕM (x)) with γx (−ϕM (x)) = y ∈ ∂Ω. In other words, for any d0 x ∈ UM there exists y ∈ ∂Ω such that x = γy (ϕM (x)) and ϕM (x) = dM (x, y) = lengthM (γy ([0, ϕM (x)])). We can then state the following result. Corollary 3.6 Under Assumption 3.1, or equivalently under the assumptions of Proposid0 d0 tion 3.5, we can define a coordinate system for UM given by T : ∂Ω × [0, d0 ) → UM such that for any (y, d) ∈ ∂Ω × [0, d0 ) we have T (y, d) = γy (d). We note that, for any 0 ≤ d < d0 , we have T (∂Ω × d) = ∂ΩdM . Moreover, if we assume that kϕM kC 1,1 (U d0 ) ≤ C0 , then T is bi-Lipschitz, that is, T and M

its inverse T −1 are Lipschitz, with Lipschitz constants bounded by a constant depending on C0 , d0 , λ, the Lipschitz constant of the metric G and C(Ω) as in (2.4) only. 25

Proof. The fact that T is injective simply depends on the uniqueness for the solution to (3.6). We begin by showing that T is Lipschitz, using an argument that is related to the continuity of solutions to ordinary differential equations with respect to the data. First of all, as for (3.3), we note that k∇M ϕM kC 0,1 (U d0 ,RN ) ≤ C1 ,

(3.7)

M

with C1 depending on C0 , λ and the Lipschitz constant of the metric G only. d0 For any i = 1, 2, let xi ∈ UM and ti ∈ [−ϕM (xi ), d0 − ϕM (xi )). We wish to estimate kγx2 (t2 ) − γx1 (t1 )k. By Volterra integral equation, we have that     Z t1 Z t2 ∇M ϕM (γx1 (s)) ds . ∇M ϕM (γx2 (s)) ds − x1 + γx2 (t2 ) − γx1 (t1 ) = x2 + 0

0

We begin by considering the case t1 = t2 . Then Z (3.8) kγx2 (t1 ) − γx1 (t1 )k ≤ kx2 − x1 k +

t1

k∇M ϕM (γx2 (s)) − ∇M ϕM (γx1 (s))k ds 0 Z t1 ≤ kx2 − x1 k + C1 kγx2 (s) − γx1 (s)k ds , 0

where we used (3.7). Then, by Gronwall lemma, we have that kγx2 (t1 ) − γx1 (t1 )k ≤ eC1 d0 kx2 − x1 k.

(3.9) Moreover, we infer that (3.10)

k(γx2 (t1 ) − x2 ) − (γx1 (t1 ) − x1 )k ≤ C1 eC1 d0 kx2 − x1 k|t1 |,

an inequality that will be crucial later on. We now turn to the general case. If t1 ≤ 0 ≤ t2 , or t2 ≤ 0 ≤ t1 , then (3.11) kγx2 (t2 ) − γx1 (t1 )k ≤ kγx2 (t2 ) − x2 k + kx2 − x1 k + kx1 − γx1 (t1 )k Z t2 Z t1 ≤ kx2 − x1 k + k∇M ϕM (γx2 (s))k ds + k∇M ϕM (γx1 (s))k ds 0 0 √ √ −1 ≤ kx2 − x1 k + λ (|t1 | + |t2 |) = kx2 − x1 k + λ−1 |t2 − t1 |, where we used (2.3) and the fact that k∇M ϕM kM = 1. Otherwise, up to swapping x1 with x2 , we have 0 ≤ t1 ≤ t2 or t2 ≤ t1 ≤ 0, and then (3.12) kγx2 (t2 ) − γx1 (t1 )k ≤ kγx2 (t2 ) − γx2 (t1 )k + kγx2 (t1 ) − γx1 (t1 )k Z t2 ≤ k∇M ϕM (γx2 (s))k ds + kγx2 (t1 ) − γx1 (t1 )k t1 √ ≤ λ−1 |t2 − t1 | + kγx2 (t1 ) − γx1 (t1 )k. By (3.9) and (3.12) we can conclude that (3.13)

kγx2 (t2 ) − γx1 (t1 )k ≤ eC1 d0 kx2 − x1 k + 26



λ−1 |t2 − t1 |.

By (3.11) and (3.13), it is immediate to prove that T is Lipschitz and that its Lipschitz constant is bounded by a constant depending on C0 , d0 , λ and the Lipschitz constant of the metric G only. d0 Let us now pass to the properties of T −1 . For any x ∈ UM , we have that T −1 (x) = (γx (−ϕM (x)), ϕM (x)) ∈ ∂Ω × [0, d0 ). We recall that ϕM is Lipschitz, with Lipschitz constant 1, with respect to the distance dM . Hence we can conclude the proof using again (3.13).  The following technical proposition is a crucial ingredient for the proof of our main decay estimate and it may be of independent interest as well. Proposition 3.7 Under Assumption 3.1, or equivalently under the assumptions of Proposition 3.5, let kϕM kC 1,1 (U d0 ) ≤ C0 . M

1,1 Let w ∈ Wloc (Ω) and let, for any 0 < d < d0 , Z S(d) = w(x) dσM (x). ∂ΩdM

We have that S is absolutely continuous on any compact subinterval of (0, d0 ) and, for almost any d, 0 < d < d0 , 0

Z

(3.14) S (d) = − ∂ΩdM

∇w(x)νM (x) dσM (x) + A(d) Z =− ∂ΩdM

where

h∇M w(x), νM (x)iM dσM (x) + A(d)

Z |A(d)| ≤ C ∂ΩdM

|w(x)| dσM (x)

for a constant C depending on C0 , d0 , λ and the Lipschitz constant of the metric G only. In particular, if w ≥ 0, then |A(d)| ≤ CS(d).

(3.15)

Remark 3.8 If w ∈ W 1,1 (Ω), then we can define Z Z S(0) = w(x) dσM (x) = ∂Ω0M

w(x) dσM (x),

∂Ω

and we have that S is absolutely continuous on any compact subinterval of [0, d0 ). 1,1 Proof. We just assume w ∈ W 1,1 (Ω) as in Remark 3.8, since, when w ∈ Wloc (Ω), the result easily follows by the arguments we present in the sequel. We begin by observing that, for any s, 0 ≤ s < d0 , we have Z Z w(x) dσM (x) = w(x)h(x) dσ(x) ∂ΩsM

∂ΩsM

27

where h(x) =

p p hG−1 (x)ν(x), ν(x)i g(x).

Moreover, for any s1 , s2 ∈ [0, d0 ), we call Ts1 ,s2 : ∂ΩsM1 → ∂ΩsM2 the change of coordinates such that Ts1 ,s2 (x) = γx (s2 − s1 ). = Ts2 ,s1 , we deduce that Ts1 ,s2 is By (3.9) and the fact that Ts1 ,s2 is invertible with Ts−1 1 ,s2 bi-Lipschitz, therefore Z Z w(γx (s2 − s1 ))h(γx (s2 − s1 ))k(x) dσ(x) w(z) dσ (z) = M s s ∂ΩM1

∂ΩM2

where k(x) can be computed as follows. For almost every x ∈ ∂ΩsM1 , with respect to the (N − 1)-dimensional Hausdorff measure, Ts1 ,s2 admits a tangential differential at x. Namely, for any orthonormal basis v1 , . . . , vN −1 of the tangent space to ∂ΩsM1 at x, there exists   ∂Ts1 ,s2 ∂Ts1 ,s2 Jτ (x) = Jτ Ts1 ,s2 (x) = (x) · · · (x) . ∂v1 ∂vN −1 Then (3.16)

k(x) =

q det ((Jτ (x))T Jτ (x)).

Let us call T˜s1 ,s2 = Ts1 ,s2 − Id and let, analogously, J˜τ (x) = Jτ T˜s1 ,s2 (x). By (3.10), we infer that for any i = 1, . . . , N − 1,



∂ T˜

s1 ,s2 (x) ≤ C1 eC1 d0 |s2 − s1 |. (3.17)

∂vi Therefore, for almost every x ∈ ∂ΩsM1 , again with respect to the (N − 1)-dimensional Hausdorff measure, we call a(x, s1 , s2 ) the number such that h(γx (s2 − s1 )) k(x) = 1 + a(x, s1 , s2 ). h(x) By using (3.16) and (3.17) to handle k(x), it is not difficult to show that, for some constant C2 depending on C0 , d0 , λ and the Lipschitz constant of the metric G only, |a(x, s1 , s2 )| ≤ C2 |s2 − s1 | for almost every x ∈ ∂ΩsM1 .

(3.18)

Then, for almost every x ∈ ∂ΩsM1 , or for almost every z = γx (s2 − s1 ) ∈ ∂ΩsM2 , Z s2 −s1 w(γx (s2 − s1 )) = w(x) + ∇w(γx (s))γx0 (s) ds 0 Z s2 −s1 Z = w(x) − ∇w(γx (s))νM (γx (s)) ds = w(x) − 0

s2 −s1

0

We call Ωs1 ,s2 the following set Ωs1 ,s2 =

 s1 s2  ΩM \ΩM

if s1 ≤ s2

ΩsM2 \ΩsM1

if s2 ≤ s1



28

∇w(γz (−s))νM (γz (−s)) ds.

and we call

 if s1 < s2  1 0 if s1 = s2 b(s1 , s2 ) =  −1 if s1 > s2 .

Then, by Fubini theorem and the coarea formula, Z

Z s

∂ΩM2

w(x) dσM (x) −

s

∂ΩM1

w(x) dσM (x)

Z =

s

∂ΩM1

w(x)a(x, s1 , s2 ) dσM (x) −

=

s

∂ΩM1

s2

dσM (z)

0

!

Z

w(x)a(x, s1 , s2 ) dσM (x) −

∂ΩtM

s1

Z =

 ∇w(γz (−s))νM (γz (−s)) ds

s

∂ΩM2

Z

Z

s2 −s1

Z

Z

∇w(x)νM (x)(1 + a(x, t, s2 )) dσM (x)

dt

Z s

∂ΩM1

∇w(x)νM (x)(1 + a(x, s, s2 )) dM (x)

w(x)a(x, s1 , s2 ) dσM (x) − b(s1 , s2 ) Ωs1 ,s2

Z =

s

∂ΩM1

w(x)a(x, s1 , s2 ) dσM (x) Z

Z

∇w(x)νM (x)a(x, s, s2 ) dM (x)

∇w(x)νM (x) dM (x) − b(s1 , s2 )

− b(s1 , s2 )

Ωs1 ,s2

Ωs1 ,s2

= A(s1 , s2 ) − B(s1 , s2 ) − C(s1 , s2 ) where, for any x ∈ Ωs1 ,s2 we set s = ϕM (x). First of all, we deduce that Z [0, d0 ) 3 s 7→ w(x) dσM (x) ∂ΩsM

is a continuous function. Again by coarea formula, we have that the function Z [0, d0 ) 3 s 7→ B(d0 /2, s) = b(d0 /2, s)

∇w(x)νM (x) dM (x) Ωd0 ,s

Z

s

!

Z

∇w(x)νM (x) dσM (x)

= d0 /2

∂ΩtM

dt

is absolutely continuous, with respect to s, on any compact subinterval of [0, d0 ) and, for almost every s1 ∈ (0, d0 ), we have B 0 (d0 /2, s1 ) = lim

s2 →s1

B(d0 /2, s2 ) − B(d0 /2, s1 ) s2 − s1 B(s1 , s2 ) = lim = s2 →s1 s2 − s1

Z s

∂ΩM1

∇w(x)νM (x) dσM (x).

The function Z [0, d0 ) 3 s 7→ D(s) =

∂ΩsM

w(x) dσM (x) + B(d0 /2, s)

29

is clearly Lipschitz continuous on any compact subinterval of [0, d0 ), therefore, for almost every s1 ∈ (0, d0 ), there exists D0 (s1 ) = lim

s2 →s1

D(s2 ) − D(s1 ) A(s1 , s2 ) − C(s1 , s2 ) = lim . s2 →s1 s2 − s1 s2 − s1

It is easy to see that C(s1 , s2 ) →0 s2 − s1 and that

as s2 → s1

Z A(s1 , s2 ) ≤ C2 |w(x)| dσM (x). s2 − s1 s ∂ΩM1

Therefore the proof can be easily concluded.  Our aim is to modify our metric G near the boundary of Ω, by multiplying it with a scalar function η, in such a way that the new metric satisfies Assumption 3.1. The construction is given in the next theorem. Theorem 3.9 Let us fix positive constants R, r and L. Let Ω ⊂ BR (0) ⊂ RN be a bounded open set of class C 1,1 with constants r and L. Let us consider d˜0 > 0 as in Theorem 2.4 and ϕ the distance to the boundary of Ω as in Definition 2.3. Let G be a Lipschitz symmetric tensor in Ω which is uniformly elliptic with constant λ, 0 < λ < 1, in Ω and such that kGkC 0,1 (Ω) ≤ C. Then there exist a constant C1 > 0, depending on r, L, R, λ and C only, and a function η ∈ C 0,1 (Ω), which is uniformly elliptic with constant λ in Ω and such that kηkC 0,1 (Ω) ≤ C1 , such that the following holds. ˜ = ηG and M ˜ the corresponding Riemannian manifold on Ω. Let ϕ ˜ be the Let us call G M d = {x ∈ Ω : ϕ (x) < d}. corresponding distance from the boundary and, for any d ≥ 0, UM ˜ ˜ M d˜ /2

˜

Then we have that U d0 /2 = UM˜0 (3.19)

and d˜ /2

in UM˜0 .

ϕM˜ = ϕ ˜

Proof. Let us define ηˆ : U d0 → R such that ˜

ηˆ = k∇M ϕk2M in U d0 . ˜

By (2.3), we obtain that λ ≤ ηˆ ≤ λ−1 in U d0 , and we have that ˜

ηˆ−1 k∇M ϕk2M = 1 in U d0 . Then we fix a cutoff function χ ∈ C ∞ (R) such that χ is decreasing, χ(t) = 1 for any t ≤ d˜0 /2 and χ(t) = 0 for any t ≥ 3d˜0 /4. We define, for any x ∈ Ω, η(x) = χ(ϕ(x))ˆ η (x) + (1 − χ(ϕ(x)) and we observe that λ ≤ η ≤ λ−1 in Ω. ˜ = ηG. By construction of η and by (2.2), we have that Let G k∇M˜ ϕkM˜ = 1 30

˜

in U d0 /2 .

Therefore, applying Proposition 3.5 with f = ϕ, we conclude that, at least in a neighbourhood of ∂Ω, ϕM˜ = ϕ. It is not difficult to show that such a neighbourhood is actually equal d˜ /2

˜

to U d0 /2 and that it coincides with UM˜0 as well. It remains to show the Lipschitz regularity of η and for this purpose it is enough to show ˜ ˜ that ηˆ is Lipschitz in U d0 . Again by (2.2), we infer that for any x ∈ U d0 ηˆ(x) = k∇M ϕ(x)k2M = h(∇ϕ(x))T , G−1 (x)(∇ϕ(x))T i. Then we can easily conclude by exploiting the Lipschitz regularity of G and the fact that ˜  ϕ ∈ C 1,1 (U d0 ) as proved in Therem 2.4. ˜ We conclude that G = ηG constructed in Theorem 3.9 is a Lipschitz symmetric tensor ˜ 0,1 in Ω which is uniformly elliptic with constant λ1 = λ2 in Ω and such that kGk C (Ω) ≤ C2 , ˜ satisfies with C2 depending on C, C1 and λ only. Moreover, by Theorem 2.4 and (3.19), G ˜ Assumption 3.1 with d0 = d0 /2.

4

The decay estimate

Let us fix positive constants R, r, L, C0 , C1 , λ and λ1 , with 0 < λ < 1 and 0 < λ1 < 1. We refer to these constants as the a priori data. Let Ω ⊂ BR (0) ⊂ RN be a bounded domain of class C 1,1 with constants r and L. Let us consider d˜0 > 0 as in Theorem 2.4 and ϕ the distance to the boundary of Ω as in Definition 2.3. Let G be a Lipschitz symmetric tensor in Ω which is uniformly elliptic with constant λ and such that kGkC 0,1 (Ω) ≤ C0 . Let A be a Lipschtitz conductivity tensor in Ω which is uniformly elliptic with constant λ1 and such that kAkC 0,1 (Ω) ≤ C1 . We further suppose that Assumption 2.10 holds. Let us fix f ∈ H 1/2 (∂Ω), with f 6= 0, and let us call Φ its frequency as in Definition 2.8. We assume that Φ > 0, that is, f is not constant on ∂Ω. Let u ∈ H 1 (Ω) be the solution to 2 (Ω) and the equation is satisfied pointwise almost everywhere (2.12). We recall that u ∈ Hloc in Ω. The important remark is that, without loss of generality, we can assume that the following fact holds. By Remark 2.13, we can assume that A = γIN with γ ∈ C 0,1 (Ω).

(4.1)

We can assume that G satisfies Assumption 3.1 with some positive constant d0 . Under this assumption, we need to add d0 and kϕM kC 1,1 (U d0 ) to the a priori data. In particular, M by Theorem 3.9 and Remark 2.12, we can assume that d0 = d˜0 /2,

(4.2)

d0 U d0 = UM

d0 and ϕM = ϕ in UM .

In this case, by Theorem 2.4, d0 and kϕM kC 1,1 (U d0 ) depend on r, L and R only. M Before stating our decay estimates, we need to set some notation. For any 0 ≤ d < d0 , let us define Z Z 2 D(d) = γ(x)k∇M u(x)kM dM (x) and H(d) = γ(x)u2 (x) dσM (x). ΩdM

∂ΩdM

31

We recall that, for any such d, ∂ΩdM = ∂(ΩdM ) and, if (4.2) holds, Ωd = ΩdM and ∂Ωd = ∂ΩdM = ∂(Ωd ). Moreover, by unique continuation, for example by [11] for N ≥ 3, and the maximum principle, both D(d) and H(d) must be strictly positive for any 0 ≤ d < d0 . We define the frequency function N as follows (4.3)

N (d) =

D(d) , H(d)

0 ≤ d < d0 .

We note that, by Remark 2.11, there exists a constant c1 , 0 < c1 < 1 depending on λ and λ1 only, such that λ1 c1 Φ ≤ N (0) ≤ (λ1 c1 )−1 Φ

(4.4)

where Φ is the frequency of the boundary datum f . For any s ≥ 0 we define  −s e if s ≤ 1 (4.5) h(s) = −1 (es) if s > 1. We note that h(0) = 1 and h is a positive C 1 strictly decreasing function. Theorem 4.1 Let f ∈ H 1/2 (∂Ω), with f 6= 0, and let its frequency Φ be positive. Under the previous assumptions and notation, there exist two positive constants C2 and c2 , depending on the a priori data only, such that, for any d, 0 < d < d0 , we have D(d) ≤ eC2 d D(0)h(c2 dΦ).

(4.6)

In the next theorem, we control the decay of the function, instead of that of its gradient. 1/2 Namely, we assume that f ∈ H∗ (∂Ω), with f 6= 0, and that Φ1 is its lower frequency. We recall that Φ1 ≤ Φ. 1/2

Theorem 4.2 Let f ∈ H∗ (∂Ω), with f 6= 0, and let Φ1 be its lower frequency. Under the previous assumptions and notation, there exist two positive constants C3 and c3 , depending on the a priori data only, such that, for any d, 0 < d < d0 /2, we have H(d) ≤ eC3 d H(0)h(c3 dΦ1 ).

(4.7)

As a corollary, we obtain a higher order decay for D with respect to the lower frequency. 1/2

Corollary 4.3 Let f ∈ H∗ (∂Ω), with f 6= 0, and let Φ1 be its lower frequency. Under the previous assumptions and notation, there exists a further absolute positive constant C4 such that, for any d, 0 < d < d0 /4, we have (4.8) D(d) ≤

C4 3C3 d/2 e H(0)h(c3 dΦ1 /2) d ≤ C4 e3C3 d/2 D(0)

h(c3 dΦ1 /2) h(c3 dΦ1 /2) ≤ C4 e3C3 d/2 D(0) . λ1 c1 dΦ λ1 c1 dΦ1

Remark 4.4 If f = φ where φ is a Steklov eigenfunction with Steklov eigenvalue µ > 0, that is, u = w where w is a solution to (1.3), then the results of Theorems 4.1 and 4.2 and of Corollary 4.3 still hold, possibly with different constants still depending on the a priori data only, even if we replace both Φ and Φ1 with the Steklov eigenvalue µ. 32

Proof of Corollary 4.3. We sketch the proof of the corollary. Let 0 < d < d0 /4. Then we have, by coarea formula and (4.7), Z 3d/2 Z 2 H(t) dt ≤ H(0)de3C3 d/2 h(c3 dΦ1 /2). γu dM = (4.9) d/2

3d/2

ΩM \ΩM

d/2

Then we apply a Caccioppoli inequality. Let χ ∈ C0∞ (R) be an even positive function such that χ is decreasing on [0, 1), χ = 1 on [0, 1/2] and χ = 0 on [3/4, +∞). We define the function ηd as follows   ϕM (x) − d ηd (x) = χ 2 for any x ∈ Ω d and we note that   2 0 ϕM (x) − d ∇M ηd (x) = χ 2 ∇M ϕM (x) d d Therefore k∇M ηd (x)kM ≤

C d

for any x ∈ Ω.

d /2

for any x ∈ Ω\ΩM0

where C is an absolute constant. Then Z 0= γh∇M u, ∇M (uηd2 )iM dM Ω Z Z 2 = γh∇M u, ∇M uiM ηd dM + 2 γh∇M u, ∇M ηd iM uηd dM . Ω



We obtain that Z Z 2 γh∇M u, ∇M uiM ηd dM = γh∇M u, ∇M uiM ηd2 dM d/2

3d/2

ΩM \ΩM



Z

Z

= −2

γh∇M u, ∇M ηd iM uηd dM = −2

d/2

3d/2

γh∇M u, ∇M ηd iM uηd dM

ΩM \ΩM



2C ≤ d and we conclude that Z (4.10)

d/2 3d/2 ΩM \ΩM

Z d/2 3d/2 ΩM \ΩM

!1/2 γh∇M u, ∇M uiM ηd2 dM

γh∇M u, ∇M uiM ηd2 dM ≤

4C 2 d2

!1/2

Z

Z d/2 3d/2 ΩM \ΩM

d/2 3d/2 ΩM \ΩM

γu2 dM

γu2 dM .

Since Z

Z 3d/4

ΩM

5d/4

γh∇M u, ∇M uiM dM ≤

\ΩM

d/2

3d/2

ΩM \ΩM

γh∇M u, ∇M uiM ηd2 dM ,

by (4.10) and (4.9), we infer that Z 4C 2 3C3 d/2 e h(c3 dΦ1 /2). (4.11) γh∇M u, ∇M uiM dM ≤ H(0) 3d/4 5d/4 d ΩM \ΩM 33

Now we consider the function ud = uηd/4 and we easily prove that Z ΩdM

Z

γk∇M ud k2M

dM ≤ 2

ΩdM

  2 γ ηd/4 k∇M uk2M + u2 k∇M ηd/4 k2M dM 16C 2 + 2 d

Z ≤2

3d/4

ΩM

5d/4

γh∇M u, ∇M uiM dM

\ΩM

!

Z 3d/4

ΩM

≤ H(0)

5d/4

γu2 dM

\ΩM

40C 2 3C3 d/2 e h(c3 dΦ1 /2). d

We have that wd = u − ud solves, in a weak sense,  divM (γ∇M wd ) = −divM (γ∇M ud ) in ΩdM wd = 0 on ∂ΩdM , from which we deduce that !1/2

Z ΩdM

γk∇M wd k2M dM

Z ≤ ΩdM

!1/2 γk∇M ud k2M dM

,

hence

160C 2 3C3 d/2 e h(c3 dΦ1 /2). d and the proof of (4.8) is concluded by taking C4 = 160C 2 . D(d) ≤ H(0)



The rest of the section is devoted to the proofs of Theorems 4.1 and 4.2. 2 (Ω), for any d with We also need the following notation. We note that, since u ∈ Hloc d 0 < d < d0 , ∇u is well-defined, in the trace sense, on ∂ΩM and that ∇u ∈ L2 (∂ΩdM , RN ). For any d with 0 < d < d0 , and almost any x ∈ ∂ΩdM , with respect to the (N − 1)dimensional Hausdorff measure, we call uνM (x) the (exterior) normal derivative of u at x with respect to ΩdM in the Riemannian setting which is given by uνM (x) = h(∇M u(x))T , νM (x)iM = ∇u(x)νM (x). We note that, analogously, uνM ∈ L2 (∂ΩdM ) is well-defined, again in the trace sense. Moreover, using the equation and the divergence theorem, we have, for any d with 0 < d < d0 , Z D(d) = γ(x)u(x)uνM (x) dσM (x). ∂ΩdM

Finally we call, for any d with 0 < d < d0 , Z T (d) = γ(x)u2νM (x) dσM (x)

and F (d) =

∂ΩdM

T (d) . D(d)

We note that, by a simple application of the Cauchy-Schwarz inequality, we have F (d) ≥ N (d)

for any 0 < d < d0 .

Following essentially the arguments developed in [11], we compute the derivatives, with respect to d, of D and H. 34

By coarea formula and the properties of ϕM , we infer that D is absolutely continuous on every compact subinterval contained in [0, d0 ) and that, for almost every d ∈ (0, d0 ), Z 0 γ(x)k∇M u(x)k2M dσM (x). D (d) = − ∂ΩdM

Then we use the following lemma, which is a suitable version of the Rellich identity. 2 (Ω), γ ∈ C 0,1 (Ω) and v = (v 1 , . . . , v N ) ∈ C 0,1 (Ω, RN ). Then Lemma 4.5 Let u ∈ Hloc

(4.12) divM (γh∇M u, ∇M uiM v) + 2divM (γ∇M u)h∇M u, viM = γh∇M u, ∇M uiM divM (v) + (∇γv)h∇M u, ∇M uiM + γ(∇g l,j v)ul uj + 2divM (γh∇M u, viM ∇M u) − 2γg l,j ul uk vjk . Proof. It follows by straightforward computations. In fact, with the summation convention and using subscripts for partial derivatives, divM (γh∇M u, ∇M uiM v) = γh∇M u, ∇M uiM divM (v) + (γg l,j ul uj )k v k = γh∇M u, ∇M uiM divM (v) + (∇γv)(g l,j ul uj ) + γ(∇g l,j v)ul uj + γg l,j (ulk uj + ul ujk )v k . On the other hand, 2divM (γh∇M u, viM ∇M u) = 2divM (γ∇M u)h∇M u, viM + 2γh∇M u, ∇(∇uv)i = 2divM (γ∇M u)h∇M u, viM + 2γg l,j ul (uk v k )j = 2divM (γ∇M u)h∇M u, viM + 2γg l,j ul uk vjk + 2γg l,j ul ukj v k . Finally, γg l,j ul ukj v k = γg l,j ul ujk v k = γg l,j ulk uj v k , which follows by symmetry of the Hessian matrix and by observing that γg l,j ulk uj v k = γg j,l ujk ul v k = γg l,j ul ujk v k since again by symmetry g l,j = g j,l . Putting the three previous equality together, the lemma is proved.  We construct a Lipschitz function v on Ω, with values in RN , coinciding with νM in By Remark 3.4 and Proposition 2.6, or with a much simpler argument if (4.2) holds, we can construct v in such a way that kvkC 0,1 (Ω,RN ) is bounded by a constant depending on the a priori data only. Then we apply the Rellich identity (4.12) in ΩdM , with 0 < d < d0 , to our solution u and such a function v. Namely, by the divergence theorem, d0 UM .

Z ∂ΩdM

Z γh∇M u, ∇M uiM dσM =

∂ΩdM

γh∇M u, ∇M uiM hv, νM iM dσM

Z (divM (γh∇M u, ∇M uiM v) + 2divM (γ∇M u)h∇M u, viM ) dM

= ΩdM

Z =2 ΩdM

Z divM (γh∇M u, viM ∇M u) dM + A0 (d) = 2

35

∂ΩdM

γu2νM dσM + A0 (d)

where Z A0 (d) =



ΩdM

 h∇M u, ∇M uiM (γdivM (v) + ∇γv) + γ(∇g l,j v)ul uj − 2γg l,j ul uk vjk dM .

In other words, for almost every d ∈ (0, d0 ), D0 (d) = −2T (d) − A0 (d). Finally, it is not difficult to show that there exists a positive constant C, depending on the a priori data only, such that for any d with 0 < d < d0 we have |A0 (d)| ≤ CD(d), consequently, for almost every d ∈ (0, d0 ), 2F (d) − C ≤ −

(4.13)

D0 (d) ≤ 2F (d) + C. D(d)

Now we turn to the computation of H 0 . We wish to prove a similar estimate, namely ˜ depending on the a priori data only, such that for that there exists a positive constant C, almost every d ∈ (0, d0 ), 0

H (d) ˜ 2N (d) − C˜ ≤ − ≤ 2N (d) + C. H(d)

(4.14)

Such a result directly follows by applying Proposition 3.7 to w = γu2 . In fact, we obtain that H is absolutely continuous on every compact subinterval contained in [0, d0 ) and that, for almost any d, 0 < d < d0 , Z 0 −H (d) = 2D(d) + (∇γ(x)νM (x))u2 (x) dσM (x) + A(d) = 2D(d) + A1 (d). ∂ΩdM

˜ depending on the Again, it is not difficult to show that there exists a positive constant C, a priori data only, such that for any d with 0 < d < d0 we have ˜ |A1 (d)| ≤ CH(d), consequently, for almost every d ∈ (0, d0 ), (4.14) holds. We are now in the position to conclude the proof of Theorem 4.1. Proof of Theorem 4.1. We use an ordinary differential equation argument, exploiting (4.13) and (4.14). With C as in (4.13), let us define, for 0 ≤ d < d0 , ˜ D(d) = e−Cd D(d). Then −

˜ 0 (d) D ≥ 2F (d) = 2(F (d) − N (d)) + 2N (d). ˜ D(d)

Therefore, for any 0 < d < d0 , ˜ ˜ log(D(0)) − log(D(d)) ≥2

Z

d

Z (F (t) − N (t)) dt + 2

0

N (t) dt. 0

36

d

We call G0 (d) = e−2

Rd 0

F

and G(d) = e−2

Rd 0

(F −N )

and G1 (d) = e−2

Rd 0

N

and we obtain that ˜ ˜ ˜ D(d) ≤ G0 (d)D(0) = G(d)G1 (d)D(0), therefore D(d) ≤ eCd G0 (d)D(0) = eCd G(d)G1 (d)D(0).

(4.15) Since

N 0 (d) D0 (d) H 0 (d) = − , N (d) D(d) H(d)

we infer that 0

(4.16)

N (d) 2(F (d) − N (d)) − Cˆ ≤ − ≤ 2(F (d) − N (d)) + Cˆ N (d)

˜ Let us define, for 0 ≤ d < d0 , where Cˆ = C + C. ˆ ˜ (d) = e−Cd N N (d).

Then, by (4.16), we conclude that −

˜ 0 (d) N ≥ 2(F (d) − N (d)) ≥ 0. ˜ (d) N

˜ is decreasing. We note that this is the crucial point in the argument of In other words, N [11]. However, in our case, such a property is not enough, since, in order to estimate G1 , we need to control how fast N can decrease. Still by (4.16), we infer that (4.17)

ˆ

N (d) ≥ e−Cd G(d)N (0).

We note that G(0) = 1 and, since F − N ≥ 0, G is positive and decreasing with respect to d. We estimate G1 (d) by using (4.17) and the fact that G(s) ≥ G(d) for any 0 < s < d, obtaining that R d −Cs ˆ ds G1 (d) ≤ e−2N (0)G(d) 0 e = e−b(d)N (0)G(d) , where

 2  ˆ −Cd 1−e . b(d) = Cˆ

We consider the auxiliary function g(x) = xe−αx , x ∈ [0, 1], with α > 0, and note that max g(x) = h(α), x∈[0,1]

thus we conclude that (4.18)

D(d) ≤ eCd D(0)h(b(d)N (0)).

37

ˆ

Since λ1 c1 Φ ≤ N (0) and 2e−Cd0 d ≤ b(d), the proof of (4.6) is concluded by setting ˆ C2 = C and c2 = 2λ1 c1 e−Cd0 .  We note that, without any control on F − N , besides the fact that it is positive, using this technique it is in practice impossible to improve the estimate of Theorem 4.1. We now turn to the proof of Theorem 4.2. We need the following notation. For any d ∈ [0, d0 ) we define Z E(d) = ΩdM

γu2 dM .

We note that E is a strictly positive function which is absolutely continuous on any compact subinterval of [0, d0 ). Moreover, for almost any d ∈ (0, d0 ), we have Z γu2 dσM = −H(d). E 0 (d) = − ∂ΩdM

d /2

We construct a Lipschitz function v1 ∈ C 0,1 (Ω, RN ) coinciding with νM in UM0 and such that kv1 kC 0,1 (Ω,RN ) is bounded by a constant depending on the a priori data only and kv1 kM ≤ 1

in Ω.

Such a construction is fairly easy. We consider a C ∞ function χ : R → R such that χ is increasing, χ = 0 on (−∞, 3d0 /5] and χ = 1 on [4d0 /5, +∞). We can define v1 with the desired properties as follows e1 v1 (x) = χ(ϕM (x)) p + (1 − χ(ϕM (x)))νM (x) hG(x)e1 , e1 i

for any x ∈ Ω.

Then we have, for any d, 0 ≤ d ≤ d0 /2, Z (4.19) H(d) = ∂ΩdM

γu2 dσM =

Z ΩdM

Z =2 ΩdM

divM (γu2 v1 ) dM Z

T

γuh(∇M u) , v1 iM dM +

ΩdM

divM (γv1 )u2 dM = 2S(d) + A2 (d).

It is not difficult to show that, for some constant C˜1 depending on the a priori data only, we have |A2 (d)| ≤ C˜1 E(d).

(4.20) We now call, for any d ∈ [0, d0 ), K(d) =

H(d) E(d)

H(d) and K1 (d) = p . E(d)

For almost any d with 0 < d < d0 , we have (4.21)



E 0 (d) = K(d). E(d)

38

Since

H 0 (d) E 0 (d) K 0 (d) = − , K(d) H(d) E(d)

by (4.14) and (4.21) we infer that, for almost every d ∈ (0, d0 ), 0

(4.22)

K (d) ˜ 2N (d) − K(d) − C˜ ≤ − ≤ 2N (d) − K(d) + C. K(d)

Analogously, since

K10 (d) H 0 (d) 1 E 0 (d) = − , K1 (d) H(d) 2 E(d)

we obtain that, for almost every d ∈ (0, d0 ), (4.23)

2N (d) −

K 0 (d) K(d) K(d) ˜ − C˜ ≤ − 1 ≤ 2N (d) − + C. 2 K1 (d) 2

We are now in the position to conclude the proof of Theorem 4.2. Proof of Theorem 4.2. In the sequel, we adopt the following normalisation, that is, we assume that (4.24)

E(0) = 1.

It is immediate to show, with this assumption, that for any d ∈ [0, d0 ) we have E(d) ≤ 1 and, consequently, K1 (d) ≤ K(d).

(4.25)

We now apply a similar technique we used before to estimate D to the function H. Namely, with C˜ as in (4.14), let us define, for any d ∈ [0, d0 ), ˜ ˜ H(d) = e−Cd H(d).

Then, by (4.25), −

  ˜ 0 (d) H K(d) K1 (d) ≥ 2N (d) ≥ 2N (d) − + . ˜ 2 2 H(d)

The main difference with respect to the argument for D is that, whereas it is immediate to show that F ≥ N , it is not that evident that 2N ≥ K/2. The other difference with respect to the previous argument is that we need to use K1 instead of K itself. However, for any d ∈ [0, d0 /2), using (4.19) and calling A˜2 (d) = A2 (d)/2, S(d) A˜2 (d) D(d) S(d) A˜2 (d) K(d) = 2N (d) − − = − − 2 E(d) E(d) E(d) S(d) + A˜2 (d) E(d) D(d)E(d) − S 2 (d) − S(d)A˜2 (d) A˜2 (d) D(d)E(d) − S 2 (d) S(d)A2 (d) A˜2 (d) =2 − =2 − − . H(d)E(d) E(d) H(d)E(d) H(d)E(d) E(d) 2N (d) −

It is easy to see that the first term M (d) = 2

D(d)E(d) − S 2 (d) H(d)E(d) 39

is positive, therefore, using (4.19) and (4.20), we obtain that   A2 (d) 1 A2 (d) K(d) = M (d) − 1− (4.26) 2N (d) − 2 E(d) 2 H(d) A2 (d) A2 (d) A22 (d) − = M1 (d) − = M (d) + 2H(d)E(d) E(d) E(d) where (4.27)

M1 (d) = M (d) +

A22 (d) ≥0 2H(d)E(d)

for any d ∈ (0, d0 /2).

We note that, by (4.20), for any d with 0 < d < d0 /2 K(d) M1 (d) − C˜1 ≤ 2N (d) − ≤ M1 (d) + C˜1 , 2

(4.28)

consequently, by (4.23) and calling C˜2 = C˜ + C˜1 , we have, for almost any d ∈ (0, d0 /2),

M1 (d) − C˜2 ≤ −

(4.29)

K10 (d) ≤ M1 (d) + C˜2 . K1 (d)

For any d with 0 < d < d0 /2, we have ˜ ˜ log(H(0)) − log(H(d)) ≥

Z 0

We call J0 (d) = e−2 and we obtain that

Rd 0

N

d

1 M1 (t) dt + 2

and J(d) = e−

Rd 0

M1

Z

d

K1 (t) dt − C˜1 d.

0

and J1 (d) = e−

Rd 0

(K1 /2)

˜ ˜ ˜ ˜ H(d) ≤ J0 (d)H(0) ≤ eC1 d J(d)J1 (d)H(0),

therefore, (4.30)

˜

˜

H(d) ≤ eCd J0 (d)H(0) ≤ eC2 d J(d)J1 (d)H(0).

For any d with 0 ≤ d < d0 /2, by (4.29) and since K1 (0) = K(0), (4.31)

˜

K1 (d) ≥ e−C2 d J(d)K(0).

We note that J(0) = 1 and, since M1 ≥ 0, J is positive and decreasing with respect to d. We estimate J1 (d) by using (4.31) and the fact that J(s) ≥ J(d) for any 0 < s < d, obtaining that R d −C˜ s ˜ 2 ds J1 (d) ≤ e−(K(0)/2)J(d) 0 e = e−b(d)K(0)J(d) , where

  ˜b(d) = 1 1 − e−C˜2 d . 2C˜2

40

Arguing as in the proof of Theorem 4.1, we conclude that, setting C3 = C˜2 , for any d with 0 < d < d0 /2 we have (4.32)

H(d) ≤ eC3 d H(0)h(˜b(d)K(0)).

In order to conclude the proof it is enough to show that, for some positive constant c3 depending on the a priori data only, we have for any d with 0 < d < d0 /2 ˜b(d)K(0) ≥ c3 dΦ1 . This is an immediate consequence of Proposition 2.17.



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