Spaces with oscillating singularities and bounded geometry

Transcription

1 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

2 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.

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

4 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 ) : Ka m (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.

5 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.)

6 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 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. For a polygon: {γ j } = { kπ α i k N } and η Ω = π α MAX.

7 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.)

8 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).

9 Cylindrical ends and oscillating conical singularities... pictures... (cylindrical ends and oscillating cylindrical ends)

10 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: 1 χ Ω T χ Ω, where T is a suitable Mellin-type integral operator (combining constructions of Schulze and Plamenevskii). 2 Multiplications with continuous functions with limits at the 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).

11 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 P ω := lim λ ω δ λ (P) := lim δ λ P δ 1 λ ω λ, P A(Ω). These limits ( limit operators RRS) correspond to the normal or indicial operators associated to a Mellin (or b-) pseudodifferential operator (Schulze, Melrose).

12 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): P ω := lim λ j ω δ λ j (P) := lim δ λ j P δ 1 λ j ω 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.

13 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),...

14 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).

15 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.

16 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.

17 Regularity and well-posedness on singular spaces Assume that: (Ω, g) is a manifold with boundary and bounded geometry. f is a bounded function on Ω such that all the covariant derivatives of log f are bounded. That is, (Ω, f 2 g, f ) is a singular manifold (H. Amann). We say that Ω satisfies the Hardy inequality if there is C > 0 s.t. Ω f 2 u 2 dvol C Ω u 2 dvol. Theorem (Ammann-Grosse-N.) With these assumptions, satisfies regularity. If, moreover, Ω satisfies the Hardy inequality, then it is well-posed.

18 Hardy inequality Let Ω be as before (i.e. (Ω, f 2 g, f ) is a singular manifold). Theorem (Bacuta-Mazzucato-N.-Zikatanov) If Ω is (contained in) a polyhedral domain and f ρ, then Ω satisfies the Hardy inequality, and hence is well-posed. Generalizes Kondratiev s well-posedness theorem: well-posedness of with Dirichlet b.c. on n-dimensional polyhedral domains (BMNZ) and on many oscillating variants à la Schulze & co. (AGN). Thank you!

19 Fredholm proof of well-posedness (Not presented in the conference.) a = is Fredholm for a ( γ 1, γ 1 ) (Kondratiev s second theorem). a = a. Hence ind( 0 ) = 0, since 0 = 0. ind( a ) = 0 for a ( γ 1, γ 1 ), by the continuity of the index. ( u, u) = Ω ( u, u)dvol, so a is injective for a 0. Hence is an isomorphism (for m = 0 and a [0, γ 1 )), hence also for a ( γ 1, 0]. For m > 0, one can use regularity. π α MAX. In particular, η Ω = γ 1 = min γ j, which, for polygons is = This is not obtained from the proof using Hardy s inequality.