REMARKS ON THE MAXIMUM PRINCIPLE FOR THE ∞-LAPLACIAN NIKOS KATZOURAKIS - JUAN MANFREDI

In this note we give three counter-examples which show that the Maximum Principle generally fails for classical solutions of a system and a single equation related to the ∞-Laplacian. The first is the tangential part of the ∞-Laplace system and the second is the scalar ∞-Laplace equation perturbed by a linear gradient term. The interpretations of the Maximum Principle for the system are that of the Convex Hull Property and also of the Maximum Principle of the modulus of the solution.

1.

Introduction

Given a smooth map u : Ω ⊆ Rn −→ RN , the ∞-Laplacian is the PDE system ∆∞ u := Du ⊗ Du + |Du|2 [Du]⊥ ⊗ I : D2 u = 0. (1) In the above, u = (u1 , . . . , uN )> , Di ≡ ∂∂xi and the gradient matrix is understood as α=1,...,N Du(x) = (Di uα (x))i=1,...,n , Entrato in redazione: 15 dicembre 2014 AMS 2010 Subject Classification: 35J47, 35J62, 49J99. Keywords: Maximum Principle, Convex Hull Property, ∞-Laplacian, Vector-valued Calculus of Variations in L∞ . The second author has been financially partially supported by NSF award DMS-1001179.

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for n, N ≥ 1. The system in index form reads 2 Di uα D j uβ D2i j uβ + |Du|2 [Du]⊥ αβ Dii uβ = 0,

where triple summation in i, j ∈ {1, . . . , n} and β ∈ {1, . . . , N} is implied although it is not explicitely written. The symbol | · | denotes the Euclidean norm on RN×n , that is |Du|2 = Di uα Di uα = Du : Du and [Du(x)]⊥ denotes the orthogonal projection in RN on the nullspace of the transpose of the linear map Du(x) : Rn −→ RN ,

x ∈ Ω.

Geometrically, [Du(x)]⊥ defines the projection on the normal space of the image of the solution at u(x). The hessian of u is viewed as a map D2 u : Ω ⊆ Rn −→ 2 RNn and D2 u(x) = (D2i j uα (x))α=1,...,N i, j=1,...,n . Because of mutual perpendicularity of the two summands comprising the ∞Laplace system, the system actually consists of a pair of two independent systems: ( Du ⊗ Du : D2 u = 0, |Du|2 [Du]⊥ ∆u = 0. The ∞-Laplace system arises as the analogue of the Euler-Lagrange equation of the supremal functional E∞ (u, Ω) := kDukL∞ (Ω) , where the L∞ norm above is interpreted as the essential supremum of the Euclidean norm of the gradient. The scalar case of ∆∞ and E∞ have a long history and first arose in the papers of Aronsson [1, 2]. Both the functional and the single equation have been extensively studied ever since, see for instance [4], [3], [6], [5] and for a pedagogical introduction see [7]. When N = 1, the ∞-Laplacian simplifies to Du ⊗ Du : D2 u = Di u D j u D2i j u = 0 and the perpendicular term |Du|2 [Du]⊥ ∆u = 0 vanishes identically. The development of the vectorial case of N ≥ 2 is much more recent and is due to the first author (see [9]-[14]). The derivation of the ∞-Laplace system from the L∞ functional and other analytical results have first been established in [9]. Almost simultaneously to [9], Sheffield and Smart studied in [18] the problem of vectorial Lipschitz extensions which amounts to changing from the Euclidean

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norm we are using herein to the operator norm on the matrix space of the gradients (this leads to a much more complicated version of “∞-Laplacian” than ours which may have even multi-valued coefficients). The vectorial case is largely under development and still not fully understood. Particular difficulties in the study of the system (1) are the emergence of highly singular solutions and that the operator definining ∆∞ may have discontinuous coefficients even when it is applied to smooth maps (see [9, 13]). The analysis of classical solutions of the system has been done mostly in [10], where there is also a maximum principle proved for ∞-Harmonic maps in 2 × 2 dimensions, namely for solutions u : Ω ⊆ R2 −→ R2 of ∆∞ u = 0. In this note we consider the question of satisfaction of the Maximum Principle for smooth solutions of the tangential part of the ∞-Laplace system when N ≥ 2 and n ≥ 1 Du ⊗ Du : D2 u = 0, u : Ω ⊆ Rn −→ RN ,

(2)

and also for smooth solutions of the single ∞-Laplace equation, perturbed by a linear smooth first order term Dv ⊗ Dv : D2 v + Dv · DF = 0, v : Ω ⊆ Rn −→ R.

(3)

For the case of the system, the Maximum Principle for the vectorial solution is considered in the sense of the standard Maximum Principle for either the projections along lines, or for the modulus of the solution. We show that in both cases, the Maximum Principle in general fails. The case of projections has an elegant restatement called the Convex Hull Property and is well-known in Calculus of Variations (see e.g. [8]): u(Ω) ⊆ co u(∂ Ω) . This inclusion says that the image is contained in the closed convex hull of the boundary values. The equivalence of this form with the form sup ξ · u ≤ max ξ · u, ∀ ξ ∈ RN , Ω

∂Ω

can be seen by writting the convex hull as the intersections of all affine halfspaces containing it. The case of the single equation is very interesting because is in direct constrast to the case of uniformly elliptic operators, where perturbation by a first order term does not violate the Maximum Principle. Our examples are the following: Example 1.1 (Failure of the Convex Hull Property for u when Du ⊗ Du : D2 u = 0). Let N ≥ 2, n ≥ 1.

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a) There exists ξ ∈ RN and u1 : Rn −→ RN in C∞ (Rn )N such that u1 solves the same system on Rn but the projection ξ · u1 along the line spanned by ξ does not satisfy neither the Maximum nor the Minimum Principle: sup ξ · u1 6≤ max ξ · u1 ∃ Ω± ⊆ Rn :

Ω+

∂ Ω+

inf ξ · u1 6≥ min ξ · u1 . − − Ω

∂Ω

b) There exists u2 : Rn \ {0} −→ RN in C∞ (Rn \ {0})N which solves Du ⊗ Du : D2 u = 0, on Rn \ {0}, such that u2 does not satisfy the Convex Hull Property over a bounded domain: ∃ Ω b Rn \ {0} : u2 (Ω) 6⊆ co u2 (∂ Ω) . Example 1.2 (Failure of the Maximum Principle for |u| when Du ⊗ Du : D2 u = 0). Let N ≥ 2, n ≥ 1. There exists u3 : Rn −→ RN in C∞ (Rn )N which solves Du ⊗ Du : D2 u = 0, on Rn , such that |u3 | does not satisfy the Maximum Principle: ∃ Ω ⊆ Rn :

sup |u3 | 6≤ max |u3 |. Ω

∂Ω

The construction of this last example can be modified in order to provide a bounded domain (along the lines of the constructions that will be made for b) of Example 1.1), but we will refrain from providing all the details in this case. The same comment applies to the Example 1.4 that follows. Corollary 1.3. The above two examples show that in general it is not possible the Maximum Principle to be true for the tangential part (2) of ∆∞ in positive codimension N − n > 0, without utilising the complete ∞-Laplace PDE system (1). In particular, it was shown in [9] that when n = 1 ≤ N, the tangential system (2) is (for classical solutions) equivalent to the poperty that the solution has constant speed. On the other hand, it was shown that the full system (1) is equivalent to that the solution is affine. Example 1.4 (Failure of the Maximum Principle for v for the single equation Dv ⊗ Dv : D2 v + Dv · DF = 0). Let n ≥ 1. There exist F ∈ C∞ (Rn ) such that the solution v ∈ C∞ (Rn ) to Dv ⊗ Dv : D2 v + Dv · DF = 0, on Rn ,

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does not satisfy neither the Maximum nor the Minimum Principle: sup v 6≤ max v, ∃ Ω± ⊆ Rn :

Ω+

∂ Ω+

inf v 6≥ min v. − − Ω

∂Ω

By taking n = 1 in the above examples, all the domains that will be exhibited become by construction bounded. Remark 1.5. We note that in the above examples the non-homogenous single ∞-Laplace equation (3) and the tangential ∞-Laplace system (2) are closely related and this fact is heavily used in our constructions: for any vector solution u of (2), after a calculation it can be seen that both the projection ξ · u along a fixed direction ξ and also the modulus |u| satisfy a non-homogeneous equation of the form (3). We conclude this introduction by noting that very recently in [15] the first author introduced a new theory of generalised solutions and proved existence to the Dirichet problem for the ∞-Laplacian. See also [16, 17].

2.

Constructions

Example 1.1. a) Let w1 ∈ Cc∞ (R) be the function 1 2 −1 (1 − t) −e , 0 < t < 2, 1 w1 (t) := 2 −1 (1 + t) +e , −2 < t < 0, 0, otherwise.

Figure 1. Illustration of the function w1 .

Fix also M > sup |w01 | R

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and define w2 ∈ C∞ (R) by taking w2 (t) :=

Z tq 0

M 2 − |w01 (s)|2 ds.

Then, the map γ : R −→ R2 given by > γ(t) := w1 (t), w2 (t) is a constant speed curve and in C∞ (R)2 : |γ 0 (t)|2 = |w01 (t)|2 + |w02 (t)|2 = M 2 , t ∈ R. Then, we define u1 : Rn −→ R2 by u1 (x) := γ(x1 ),

x = (x1 , . . . , xn )> ∈ Rn , n ≥ 1.

The smooth map u1 is a classical global solution of the tangential part of the ∞-Laplacian: since > Du1 (x) = γ 0 (x1 ), 0, . . . , 0 ,

|Du1 (x)|2 = |γ 0 (x1 )|2 ,

we have

1 1 2 Du (x) ⊗ Du (x) : D u (x) = Du (x) D |Du | (x) 2 1 02 0 0 = γ (x1 ) |γ | (x1 ) 2 = 0, 1

1

2 1

1

for all x ∈ Rn . However, for Ω− = (−2, 0) × R, Ω+ = (0, 2) × R, and for ξ = e1 = (1, 0)> , we have sup ξ · u1 > max ξ · u1 = 0 = min ξ · u1 > inf ξ · u1 . − − Ω+

∂ Ω+

∂Ω

Ω

Hence, ξ · u1 does not satisfy neither the maximum nor the minimum principle. By writing a convex set as the intersection of all halfspaces containing it, it

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69

follows that u1 can not satisfy the convex hull property either.

Figure 2. Illustration of the Convex Hull Property.

b) We can also modify the previous construction of a) in order to have bounded domains Ω± . Indeed, define 1 2 −1 (2 − t) , 1 < t < 3, e z1 (t) := 0, otherwise, and choose as before M > sup |z01 |, R

z2 (t) :=

Z tq 0

M 2 − |z01 (s)|2 ds.

Then, we define u2 : Rn \ {0} −→ R2 in C∞ (Rn \ {0})2 by taking > u2 (x) := z1 (|x|), z2 (|x|) , x ∈ Rn \ {0}, n ≥ 1. Then, u2 is a smooth solution of the vectorial Eikonal equation: indeed, since Du2 (x) = z01 (|x|), z02 (|x|))> ⊗

x , x 6= 0, |x|

we have 2 2 2 2 0 Du (x) = z1 (|x|) x + z02 (|x|) x |x| |x| 0 2 0 2 = z1 (|x|) + z2 (|x|) 2 2 = z01 (|x|) + M 2 − z01 (|x|) = M2,

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for any x ∈ Rn \ {0}. Hence, u2 is a classical solution of the tangential part of the ∞-Laplacian: 1 2 2 2 2 2 2 2 Du ⊗ Du : D u = Du D = 0. Du 2 However, for Ω := {x ∈ Rn : 1 < |x| < 3} and ξ := e1 = (1, 0)> , we have sup ξ · u2 > max ξ · u2 = 0. ∂Ω

Ω

Hence, the Convex Hull Property fails for u2 on Ω. Example 1.2. Suppose u : Rn −→ RN is in C∞ (Rn )N and satisfies |u| > 0. Then, we write in polar coordinates u = ρ n, ρ := |u|, n :=

u , |u|

and calculate Du = n ⊗ Dρ + ρDn. Since |n|2 = 1, we have by differentation n> Dn = 0, that is n ⊥ Di n, i = 1, . . . , n. Hence, by perpendicularity |Du|2 = |Dρ|2 + ρ 2 |Dn|2 . Then, we may also calculate

1 2 Du ⊗ Du : D u = Du D |Du| 2 1 = n ⊗ Dρ + ρDn D |Dρ|2 + ρ 2 |Dn|2 . 2 2

We now define functions ρ ∗ ∈ C∞ (Rn ), n∗ ∈ C∞ (Rn )N with ρ ∗ > 0 on Rn , |n∗ |2 ≡ 1 on Rn ,

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and show that if we define u3 by taking ρ ∗ and n∗ as its polar and radial coordinates, then u3 is a global solution of (2) on Rn , but |u3 | fails to satisfy the Maximum Principle. Indeed, we define 2

ρ ∗ (t) := e−|t|

Figure 3. Illustration of the function ρ ∗ .

and fix a constant M > sup |(ρ ∗ )0 |. R

Let now γ : R −→ e.g. we can take

R2

be a smooth unit speed curve parameterising the circle, γ(t) := (cost, sint)> .

Then, we define Z t

K(t) := 0

s

2 M 2 − (ρ ∗ )0 (s) ds (ρ ∗ )2 (s)

and n∗ (t) := γ K(t) . The previous definitions imply that ρ ∗ , n∗ satisfy (n∗ )0 (t) = γ 0 K(t) K 0 (t) and hence |(n∗ )0 (t)| = γ 0 K(t) K 0 (t) = K 0 (t) , while, 2 − (ρ ∗ )0 (t) 2 0 2 M K (t) = (ρ ∗ )2 (t) which (by combining the previous two equalities) implies 2 2 (ρ ∗ )2 (t) (n∗ )0 (t) + (ρ ∗ )0 (t) = M 2 , for all t ∈ R. Hence, if

u3 : Rn −→ R2

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is given by u3 (x) := ρ ∗ (x1 )n∗ (x1 ), x = (x1 , . . . , xn )> ∈ Rn , we have that u3 is smooth and since |Du3 | ≡ M, it solves Du3 ⊗ Du3 : D2 u3 = 0, on Rn . However, for Ω := x ∈ Rn : |x1 | < 1 , we have that |u3 | = ρ ∗ does not satisfy the maximum principle: sup |u3 | = 1 > Ω

1 = max |u3 |. e ∂Ω

Example 1.4. In the setting of Example 1.1, consider again the function w1 ∈ Cc∞ (R) given by 1 2 − e (1 − t) − 1 , 0 < t < 2, 1 w1 (t) := 2 + e (1 + t) − 1 , −2 < t < 0, 0, otherwise. Fix M > sup |w01 | R

and define w2

∈ C∞ (R)

by taking

w2 (t) :=

Z tq 0

M 2 − |w01 (s)|2 ds.

Then, the map γ : R −→ R2 given by > γ(t) := w1 (t), w2 (t) is a constant speed curve and in C∞ (R)2 : |γ 0 (t)|2 = |w01 (t)|2 + |w02 (t)|2 = M 2 , t ∈ R. Then, we define v : Rn −→ R in C∞ (Rn ) by v(x) := w1 (x1 ),

x = (x1 , . . . , xn )> ∈ Rn , n ≥ 1.

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We also define F : Rn −→ R in C∞ (Rn ) by F(x) :=

2 1 0 w2 (x1 ) , 2

x = (x1 , . . . , xn ) ∈ Rn , n ≥ 1.

Given v, F as above, we calculate 1 0 2 0 w Dv ⊗ Dv : D v + Dv · DF = + 2 2 1 2 0 1 0 2 0 2 0 + w1 M − w1 = w01 w01 2 2 = 0, w01 w01 w001

2

w01

on Rn . Hence, v is a global smooth solution of Dv ⊗ Dv : D2 v + Dv · DF = 0, on Rn . However, for Ω− := (−2, 0) × R, Ω+ := (0, 2) × R, we have sup v > max v = 0 = min v > inf v. − − Ω+

∂ Ω+

∂Ω

Ω

Hence, v does not satisfy neither the Maximum nor the Minimum Principle.

REFERENCES [1] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv f¨ur Mat. 6 (1967), 551–561. [2] G. Aronsson, On the partial differential equation u2x uxx + 2ux uy uxy + u2y uyy = 0, Arkiv f¨ur Mat. 7 (1968), 395–425. [3] E. N. Barron - L. C. Evans - R. Jensen, The Infinity Laplacian, Aronsson’s Equation and their Generalizations, Transactions of the AMS 360 (1) (2008). [4] M. G. Crandall, A visit with the ∞-Laplacian, in Calculus of Variations and NonLinear Partial Differential Equations, Springer Lecture notes in Mathematics 1927, CIME, Cetraro Italy 2005. [5] M. G. Crandall - G. Gunnarsson - P. Y. Wang, Uniqueness of ∞-harmonic Functions and the Eikonal Equation, Comm. Partial Differential Equations 32 (10-12) (2007), 1587–1615.

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[6] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis 17, VIII 1993, Volume 123, Issue 1, 51–74. [7] N. Katzourakis, An introduction to viscosity solutions for fully nonlinear PDE with applications to Calculus of Variations in L∞ , Springer Brief in Mathematics, DOI 10.1007/978-3-319-12829-0, ISBN: 978-3-319-12828-3, 2014-15. [8] N. Katzourakis, Maximum Principles for Vectorial Approximate Minimizers of Nonconvex Functionals, Calculus of Variations and PDE, DOI 10.1007/s00526012-0491-6, published online (2012). [9] N. Katzourakis, L∞ Variational problems for maps and the Aronsson PDE system, J. Differential Equations 253 (7) (2012), 2123–2139. [10] N. Katzourakis, On the Structure of ∞-Harmonic Maps, Communications in PDE 39 (11) (2014), 2091–2124. [11] N. Katzourakis, ∞-Minimal Submanifolds, Proceedings of the Amer. Math. Soc. 142 (2014), 2797–2811. [12] N. Katzourakis, Optimal ∞-Quasiconformal Immersions, ESAIM Control, Opt. and Calc. Var., to appear (2015) DOI: http://dx.doi.org/10.1051/cocv/2014038. [13] N. Katzourakis, Explicit 2D ∞-harmonic maps whose interfaces have junctions and corners, Comptes Rendus Acad. Sci. Paris Ser.I 351 (2013), 677–680. [14] N. Katzourakis, Nonuniqueness in vector-valued Calculus of Variations in L∞ and some linear elliptic systems, Communications on Pure and Applied Analysis 14 (1) (2015), 313–327. [15] N. Katzourakis, Generalised Solutions for Fully Nonlinear PDE Systems and Existence-Uniqueness Theorems, ArXiv preprint 2015. [16] N. Katzourakis, Existence of Generalised Solutions to the Equations of vectorial Calculus of Variations in L∞ , ArXiv preprint 2015. [17] N. Katzourakis, Equivalence between weak and D-solutions for Symmetric Hyperbolic PDE Systems, ArXiv preprint 2015. [18] S. Sheffield - C. K. Smart, Vector Valued Optimal Lipschitz Extensions, Comm. Pure Appl. Math. 65 (1) (2012), 128–154.

NIKOS KATZOURAKIS Department of Mathematics University of Reading, UK e-mail: [email protected] JUAN MANFREDI Department of Mathematics University of Pittsburgh, USA e-mail: [email protected]