# Singular Elliptic Partial Differential Equations

manifolds with a Lie structure at in nity, Ann. Math. (2) 165, No. 3, 717-747 (2007). (construction of a ’small’ algebra of pseudodi erential operator...

Singular Elliptic Partial Differential Equations Daniel Grieser (Carl von Ossietzky Universit¨ at Oldenburg)

September 19, 2012

Summer School ’Singular Analysis’

Daniel Grieser (Oldenburg)

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Singular elliptic PDE: Outline 1

Problems Typical problems (Uniform) ellipticity and degeneracy Non-compact = singular Types of degeneration via vector fields: Examples General setup for singular elliptic problems

2

Methods Model problems and pseudodifferential calculus The Schwartz kernel

3

Who did what? (and named it how?)

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References

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Singular PDE: Problems Consider linear, elliptic operators with smooth coefficients P on non-compact manifold M, with compactification M Pε on Mε , ε ∈ (0, 1], ’degenerating’ as ε → 0 (+ boundary conditions if needed)

Typical Problems Asymptotics (or regularity) of solutions of Pε u = f : u(p, ε) ∼??? as p → ∂M or ε → 0 or both Asymptotics of eigenvalues, eigenfunctions of Pε (as ε → 0) Master problem: Understand the resolvent kernel: Asymptotics of (Pε − z)−1 (p, p 0 ) as function of p, p 0 , z, ε.

Daniel Grieser (Oldenburg)

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(Uniform) ellipticity and degeneracy In P = σ(p, ∂p ), replace ∂p by ξ. Let m = order of P, σm = principal part. P elliptic: |σm (p, ξ)| ≥ c(p)|ξ|m , c(p) > 0. P uniformly elliptic: c(p) ≥ c0 > 0. Example: P = (x∂x )2 + ∂y2 has σ2 = x 2 ξ 2 + η 2 , not uniformly elliptic in x > 0. no information on behavior as x → 0 But: P = b σ(x∂x , ∂y ) where b σ(τ, η) = τ 2 + η 2 is uniformly elliptic symbol! Note: 2 P = -Laplace-Operator of metric dx + dy 2 (cylinder) x x −2 P = -Laplace-Operator (+ l.o.t.) of metric dx 2 + x 2 dy 2 (cone) The vector fields x∂x , ∂y define a type of degeneration (or singularity). Daniel Grieser (Oldenburg)

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Non-compact problems = singular problems Example Set r = x1 , then r → ∞ corresponds to x → 0 and ∂r for r ∈ (1, ∞)

=

−x 2 ∂x for x ∈ (0, 1)

Convention Put special behavior (singular, non-compact) at x = 0.

Example Compactify Rn by using polar coordinates (r , ω), setting r = the sphere at infinity x = 0. Metric: 

dx x2

2

 +

dω x

1 x

2

(dω)2 = metric on unit sphere Daniel Grieser (Oldenburg)

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Types of degeneration via vector fields: Examples Singularity type smooth, compact

Vector fields (local basis) ∂xi

infinite cylinder, cone near its tip

x∂x , ∂yi

(x > 0)

(also: Coulomb potential)

cone near infinity

x 2 ∂x , x∂yi

edge

x∂x , x∂yi , ∂zj

fibred cusp (ex.: loc.symm.space) adiabatic/semiclassical limit adiabatic limit with ends adiabatic limit with conical singularities (e.g. thin triangles)

x 2 ∂x , x∂yi , ∂zj ε∂yi , ∂zj ... ...

AND SO ON ... Daniel Grieser (Oldenburg)

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General setup for singular elliptic problems Given: X : a compact manifold with boundary (or corners) V : a Lie algebra of vector fields on X (locally free C ∞ (X ) module) This defines notions of Diff m V (X ) and V-ellipticity.

Goals: Construct parametrices of V-elliptic elements of Diff m V (X ), up to remainder which is smoothing compact rapidly vanishing (at the boundary, or at least some faces)

Classical case X has no boundary, V = all smooth vector fields on X R ∈ Ψ−∞ (X ) ⇒ R smoothing ⇒ R compact Daniel Grieser (Oldenburg)

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Methods Constructive approach 1

Solve model problems (= limit problems)

2

Patch model solutions together

3

Justify: Show that we get approximate solution; remove/estimate errors

Non-singular case: Invert uniformly elliptic operator P = p(p, ∂p ) 1 Model problems: P p0 = pm (p0 , ∂p ), x0 ∈ M (’zoom in’ at x0 ) (constant coefficients invert by Fourier transform, get Qp0 (p, p 0 )) 2 Patch: Q(p, p 0 ) := Q (p, p 0 ) p 3 Justify: PQ = I + R, ord(R) = −1 Pseudodifferential calculus! (composition of ΨDOs, short exact symbol sequence) Case of conical singularity: Additional model problem at tip of cone, solved by Mellin transform. cone (or b-) calculus Daniel Grieser (Oldenburg)

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The Schwartz kernel Schwartz kernel KA ∈ D0 (M 2 ) of a linear operator A on C ∞ (M): Z (Af )(p) = KA (p, p 0 ) f (p 0 ) dp 0 M

’Knowing’ KP −1 yields info. on solutions of Pu = f , e.g. (supp f compact): P = ∆ on Rn , n ≥ 3: KP −1 (p, p 0 ) = cn |p − p 0 |2−n . ’Order -2’ singularity at diagonal p = p 0 ⇒ regularity of u Polynomial decay as |p − p 0 | → ∞ ⇒ polynomial decay of u at ∞ ˆ − p 0 ), h(ξ) = P = ∆ + 1: KP −1 (p, p 0 ) = h(p

1 . |ξ|2 +1

’Order -2’ singularity at diagonal p = p 0 ⇒ regularity of u Exponential decay as |p − p 0 | → ∞ ⇒ rapid decay of u at ∞ Daniel Grieser (Oldenburg)

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Some singular ΨDO calculi Geometry cone

Major originators Schulze (cone calculus) Melrose (b-calculus) cone near ∞ x 2 ∂x , x∂yi Schulze (exit to infinity) Melrose (scattering calculus) edge x∂x , x∂yi , ∂zj Schulze Mazzeo polyhedron Schulze Albin-Leichtnam-Mazzeo-Piaz fibred cusp x 2 ∂x , x∂yi , ∂zj Mazzeo-Melrose (ϕ-calc.) Vaillant, G.-Hunsicker adiab./semiclass. limit ε∂yi , ∂zj Mazzeo-Melrose, . . . adiab. limit with ends Hassell-Mazzeo-Melrose adiab. limit with cone sing. G.-Melrose (in progress) Any ’boundary fibration structure’ Melrose (a general framework, in progress) Daniel Grieser (Oldenburg)

Vector fields x∂x , ∂yi

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Types of results Fredholm-property on weighted Sobolev spaces Asymptotics of solutions Asymptotics of spectral invariants and eigenfunctions Scattering theory: Analytic continuation of the resolvent, construction of generalized eigenfunctions, description of scattering matrix Heat kernel asymptotics Propagation of singularities for wave equation Applications to global analysis, differential geometry etc. (Everything on certain classes of singular spaces, to varying precision.) Wide open problem: Do it for any ’reasonable’ (e.g. algebraic) singularity! Daniel Grieser (Oldenburg)

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Some references I



P. Albin and E. Leichtnam and R. Mazzeo and P. Paolo, The signature ´ Norm. Sup´er. (4) 45, No. 2, 241-310 package on Witt spaces, Ann. Sci. Ec. (2012). (includes analysis on spaces with iterated conical/edge singularities)



B. Ammann and R. Lauter and V. Nistor, Pseudo-differential operators on manifolds with a Lie structure at infinity, Ann. Math. (2) 165, No. 3, 717-747 (2007). (construction of a ’small’ algebra of pseudodifferential operators in a fairly general context by Lie groupoid/algebroid methods)

Daniel Grieser (Oldenburg)

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Some references II 

D.Grieser, Basics of the b-calculus, in J.B.Gil et al. (eds.), Approaches to Singular Analysis, 30-84, Operator Theory: Advances and Applications, 125. Advances in Partial Differential Equations, Birkh¨auser, Basel, 2001. http://www.staff.uni-oldenburg.de/daniel.grieser/wwwpapers/ GriBBC.pdf (leisurely elementary introduction to manifolds with corners, blow-ups and the b-calculus)



D. Grieser, E. Hunsicker, Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I, Journal of Functional Analysis, 2009. (generalizes  to the case of several stacked fibrations)



R. Mazzeo, Elliptic theory of differential edge operators I, Comm. Part. Diff. Eq., 16 (1991) no. 10, 1616-1664. (ΨDO calculus for manifolds with edges: x∂x , x∂y , ∂z )

Daniel Grieser (Oldenburg)

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Some references III 

R. Mazzeo and R. Melrose, Pseudodifferential operators on manifolds with fibred boundaries in “Mikio Sato: a great Japanese mathematician of the twentieth century.”, Asian J. Math. 2 (1998) no. 4, 833–866. (small ΨDO calculus for fibred cusp operators: x 2 ∂x , x∂y , ∂z )



R.B.Melrose, Pseudodifferential operators, corners and singular limits, Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 217-234 (1991). (introduction of a general framework for singular analysis, with examples)



R. Melrose, Differential analysis on manifolds with corners, in preparation, partially available at http://www-math.mit.edu/~rbm/book.html. (the details for , work in progress)



R. Melrose, The Atiyah-Patodi-Singer index theorem, A.K. Peters, Newton (1991). (detailed introduction of the b-ΨDO calculus, x∂x , ∂y – elliptic and heat kernel parametrix – and application to index theory) Daniel Grieser (Oldenburg)

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Some references IV

 B-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, John Wiley & Sons, (2008). (ΨDO calculus for cone and edge singularities, including boundary value problems)  B. Vaillant, Index and spectral theory for manifolds with generalized fibred cusps, Ph.D. thesis, Univ. of Bonn, 2001. arXiv:math-DG/0102072. (extends the parametrix construction of  to the case of non-invertible normal operator; also heat kernel and application to index theory)

Daniel Grieser (Oldenburg)

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