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

logo-lorraine Kondratiev’s well-posedness and Fredholm theorems Oscillating conical points and Fredholm operators Bounded geometry Bonus:

Spaces with ‘oscillating singularities’ and bounded geometry

Victor Nistor1

1Université 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

1

Kondratiev’s well-posedness and Fredholm theorems

2

Oscillating conical points and Fredholm operators

3

Bounded geometry

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Bonus: Kondratiev and index theory Collaborators: B. Ammann, C. Carvalho, N. Grosse, A. Mazzucato, M. Kohr, Yu Qiao, A. Weinstein, P . Xu.

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Kondratiev’s spaces

Ω ⊂ M =bounded domain, M =Riemannian manifold. ∂singΩ ⊂ ∂Ω is the set of singular boundary points of Ω ρ(x) := dist(x, ∂singΩ). Kondratiev’s weighted Sobolev spaces (M = Rn) : Km

a (Ω) := { u | ρ|α|−a∂αu ∈ L2(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 ηΩ > 0 such that, for all m ∈ Z+ and |a| < ηΩ, we have an isomorphism ∆a = ∆ : Km+1

a+1 (Ω) ∩ {u|∂Ω = 0} → Km−1 a−1 (Ω) .

It reduces to a well-known, classical result if Ω is smooth. (The a is a new feature.)

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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 < γj ր ∞ such that ∆a = ∆ : Km+1

a+1 (Ω) ∩ {u|∂Ω = 0} → Km−1 a−1 (Ω)

is Fredholm if, and only if, a = ±γj. Moreover, ηΩ = γ1 = min γj, which is not obtained from the alternative proof using Hardy’s inequality. For a polygon: {γj} = kπ

αi | k ∈ N

• and ηΩ =

π αMAX .

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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

• n singular spaces (cusps). (Also: Debord-Skandalis,

Melrose, N.-Weinstein-Xu.)

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Rabinovich-Schulze-Tarkhanov: oscillating conical pts

Typically for Fredholm conditions: “nice ends.” Examples: (asymptotically) cylindrical, conical, euclidean,

• r 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).”

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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 ⊂ Rn and is based at 0. The algebra A(Ω) considered by RST is the norm closed algebra generated by:

1

χΩTχΩ, 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).

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Limit operators

Assume Ω is based at 0. Let δλ =dilation by λ > 0 on Rn ⊃ Ω. If the cone is “straight” and ω ∈ {0, ∞}, we have limits Pω := lim

λ→ω δλ(P) := lim λ→ω δλ ◦ P ◦ δ−1 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).

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Fredholm conditions

In general, the limits defining the limit operators will exist

• nly for suitable subsequences λj.

In fact, they exist for λj → ω, where ω belongs to a suitable compactification of Ω. The limit operators associated to P ∈ A (RRS, RST): Pω := lim

λj→ω δλj(P) :=

lim

λj→ω δλj ◦ P ◦ δ−1 λj .

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.

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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), ...

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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

• rder 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).

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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, ∆ : Hm+1(M) ∩ {u = 0 on A and ∂νu = 0 on Ac } → Hm−1(M) is an isomorphism. We say that (M, A) has finite width.

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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, (∆, Bx) 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 Cm ≥ 0 such that uHm+1 ≤ Cm

• ∆uHm−1 + BuHm+1/2−j + uH1
• .

In particular, the Dirichlet and Neumann boundary conditions satisfy the uniform Shapiro-Lopatinski regularity condition.

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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 −2g, 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|2dvol ≤ C
• Ω |∇u|2dvol.

Theorem (Ammann-Grosse-N.) With these assumptions, ∆ satisfies regularity. If, moreover, Ω satisfies the Hardy inequality, then it is well-posed.

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Hardy inequality

Let Ω be as before (i.e. (Ω, f −2g, 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.

• n n-dimensional polyhedral domains (BMNZ) and
• n many oscillating variants à la Schulze & co. (AGN).

Thank you!

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“Fredholm proof” of “well-posedness”

(Not presented in the conference.) ∆a = ∆ is Fredholm for a ∈ (−γ1, γ1) (Kondratiev’s second theorem). ∆∗

a = ∆−a.

Hence ind(∆0) = 0, since ∆∗

0 = ∆0.

ind(∆a) = 0 for a ∈ (−γ1, γ1), 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, γ1)), hence also for a ∈ (−γ1, 0]. For m > 0, one can use regularity. In particular, ηΩ = γ1 = min γj, which, for polygons is =

π αMAX .

This is not obtained from the proof using Hardy’s inequality.