Spaces with oscillating singularities and bounded geometry Victor - PowerPoint PPT Presentation

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Spaces with ‘oscillating singularities’ and bounded geometry Victor Nistor 1 1 Université de Lorraine (Metz), France Potsdam, March 2019, Conference in Honor of B.-W. Schulze logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: ABSTRACT Rabinovich, Schulze and Tarkhanov (RTS): domains with “oscillating singularities.” Oscillating conical: one replaces the asymp. straight cylindrical end (Kondratiev) with an oscillating one. New feature : new characterization of the Fredholm property (generalizing Kondratiev’s Fredholm conditions). My talk: I will review some of these results, ... ... and then I will discuss their relation to manifolds with boundary and bounded geometry and to results of H. Amann and of myself jointly with Ammann and Grosse. logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Summary Kondratiev’s well-posedness and Fredholm theorems 1 Oscillating conical points and Fredholm operators 2 Bounded geometry 3 Bonus: Kondratiev and index theory 4 Collaborators: B. Ammann, C. Carvalho, N. Grosse, A. Mazzucato, M. Kohr, Yu Qiao, A. Weinstein, P . Xu. logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Kondratiev’s spaces Ω ⊂ M =bounded domain, M =Riemannian manifold. ∂ sing Ω ⊂ ∂ Ω is the set of singular boundary points of Ω ρ ( x ) := dist ( x , ∂ sing Ω) . Kondratiev’s weighted Sobolev spaces ( M = R n ) : K m a (Ω) := { u | ρ | α |− a ∂ α u ∈ L 2 (Ω) , | α | ≤ m } . If ∂ Ω is smooth: ρ = 1 and usual spaces. Schulze includes sometimes singular functions. logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Kondratiev’s well-posedness theorem Kondratiev’s results are for domains with conical points . Theorem (Kondratiev ’67, Kozlov-Mazya-Rossmann) Let Ω be a bounded domain with conical points. Then there exists η Ω > 0 such that, for all m ∈ Z + and | a | < η Ω , we have an isomorphism ∆ a = ∆ : K m + 1 a + 1 (Ω) ∩ { u | ∂ Ω = 0 } → K m − 1 a − 1 (Ω) . It reduces to a well-known, classical result if Ω is smooth. (The a is a new feature.) logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Kondratiev’s Fredholm alternative for conical points Kondratiev’s “proof” of his well-posedness theorem: using Fredholm operators ( Ω bounded with conical points). Theorem (Kondratiev ’67) There is 0 < γ j ր ∞ such that ∆ a = ∆ : K m + 1 a + 1 (Ω) ∩ { u | ∂ Ω = 0 } → K m − 1 a − 1 (Ω) is Fredholm if, and only if, a � = ± γ j . Moreover, η Ω = γ 1 = min γ j , which is not obtained from the alternative proof using Hardy’s inequality. � k π π � For a polygon: { γ j } = α i | k ∈ N and η Ω = α MAX . logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Pseudodifferential operators Underscores the importance of Fredholm conditions. A convenient approach: via pseudodifferential operators . Many contributions by Schulze and his collaborators, as well as by many other people: Brunning, Krainer, Lesch, Melrose, Mendoza, Rabinovich, Roch, Schrohe, Vasy, ... Lauter and Seiler: nice paper in which they describe the differences between the approaches. Schulze-Sternin-Shatalov: the role of Lie algebras of vector fields in understanding pseudodifferential operators on singular spaces (cusps). (Also: Debord-Skandalis, Melrose, N.-Weinstein-Xu.) logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Rabinovich-Schulze-Tarkhanov: oscillating conical pts Typically for Fredholm conditions: “nice ends.” Examples: (asymptotically) cylindrical, conical, euclidean, or hyperbolic spaces. “Nice ends” often means the existence of a compactification to a manifold with corners . This is not the case for “oscillating conical points (pts).” logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Cylindrical ends and oscillating conical singularities .... pictures .... (cylindrical ends and oscillating cylindrical ends) logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: The algebra A (Ω) NEW Assume Ω ⊂ R n and is based at 0. The algebra A (Ω) considered by RST is the norm closed algebra generated by: χ Ω T χ Ω , where T is a suitable Mellin-type integral operator 1 (combining constructions of Schulze and Plamenevskii). Multiplications with continuous functions with limits at the 2 “infinities” of the cone (0 and ∞ ). RST: characterization of Fredholm operators in A (Ω) using “limit operators” (Rabinovich-Roch-Silbermann) and Simonenko’s local principle. The limit operators are obtained via dilations (next). logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Limit operators Assume Ω is based at 0. Let δ λ = dilation by λ > 0 on R n ⊃ Ω . If the cone is “straight” and ω ∈ { 0 , ∞} , we have limits λ → ω δ λ ◦ P ◦ δ − 1 P ω := lim λ → ω δ λ ( P ) := lim , P ∈ A (Ω) . λ These limits ( “limit operators” RRS) correspond to the “normal” or “indicial” operators associated to a Mellin (or b -) pseudodifferential operator (Schulze, Melrose). logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Fredholm conditions In general, the limits defining the limit operators will exist only for suitable subsequences λ j . In fact, they exist for λ j → ω, where ω belongs to a suitable compactification of Ω . The limit operators associated to P ∈ A (RRS, RST): λ j → ω δ λ j ◦ P ◦ δ − 1 P ω := λ j → ω δ λ j ( P ) := lim lim λ j . Theorem (Rabinovich-Schulze-Tarkhanov (RST)) An operator P ∈ A (Ω) is Fredholm if, and only if, it is elliptic and all its limit operators are invertible. logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Comments The “ellipticity” refers to the invertibility of certain symbols associated to points of Ω , with boundary points contributing a “non-commutative symbol,” à la Plamenevskii, whereas the interior points contributing the usual principal symbol. The exactly periodic (oscillating) case was recently studied by S. Melo (no boundaries). Many similar results in a QM framework, but nice ends and again no boundaries: Côme, Georgescu, Mantoiu, Mougel, Purice, Richard, Carvalho-N.-Qiao (that’s how I got to be interested in the RST result), ... logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Evolution equations and Amann’s “singular manifolds” Hyperbolic equations do not “see” the ends (they don’t care if the ends are nice or not): finite propagation speed. Several maximal regularity results by H. Amann (second order equations: Krainer, Mazzucato-N.). H. Amann: a framework to study PDEs on manifolds with boundary and bounded geometry (Schick) together with a conformal weight factor ( “singular manifolds” ). The manifolds with oscillating conical points are wonderful (non-polyhedral) examples of singular manifolds (the weight is ρ =the distance to the singular points). logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Well-posedness for mixed boundary value problems Next , v. brief account of some results in the bounded geometry and “singular manifolds” settings ( ∆ for simplicity). In what follows, M will be a manifold with boundary and bounded geometry. Theorem (Ammann-Grosse-N.) Let A ⊂ ∂ M be a union of connected components such that dist ( x , A ) is bounded on M. Then, for all m ∈ N , ∆ : H m + 1 ( M ) ∩ { u = 0 on A and ∂ ν u = 0 on A c } → H m − 1 ( M ) is an isomorphism. We say that ( M , A ) has finite width . logo-lorraine

Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus: Regularity and bounded geometry We consider a boundary operator B and we assume that (∆ , B ) satisfies a uniform Shapiro-Lopatinski regularity condition . That is, at each point x of the boundary, (∆ , B x ) satisfies the Shapiro-Lopatinski condition with bounds independent of x . We then have the following regularity result: Theorem (Grosse-N.) For all m ∈ N , there exists C m ≥ 0 such that � � � u � H m + 1 ≤ C m � ∆ u � H m − 1 + � Bu � H m + 1 / 2 − j + � u � H 1 . In particular, the Dirichlet and Neumann boundary conditions satisfy the uniform Shapiro-Lopatinski regularity condition. logo-lorraine