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Exponential stabilization of waves for the Zaremba boundary condition

Pierre Cornilleau, Luc Robbiano

Abstract

In this article we prove, under some geometrical condition on geodesic flow, exponential stabilization of wave equation with Zaremba boundary condition. We prove an estimate on the resolvent of semigroup associated with wave equation on the imaginary axis and we deduce the stabilization result. To prove this estimate we apply semiclassical measure technics. The main difficulties are to prove that support of measure is in characteristic set in a neighborhood of the jump in the boundary condition and to prove results of propagation in a neighborhood of a boundary point where Neumann boundary condition is imposed. In fact if a lot of results applied here are proved in previous articles, these two points are new.

Exponential stabilization of waves for the Zaremba boundary condition

Abstract

In this article we prove, under some geometrical condition on geodesic flow, exponential stabilization of wave equation with Zaremba boundary condition. We prove an estimate on the resolvent of semigroup associated with wave equation on the imaginary axis and we deduce the stabilization result. To prove this estimate we apply semiclassical measure technics. The main difficulties are to prove that support of measure is in characteristic set in a neighborhood of the jump in the boundary condition and to prove results of propagation in a neighborhood of a boundary point where Neumann boundary condition is imposed. In fact if a lot of results applied here are proved in previous articles, these two points are new.

Paper Structure

This paper contains 45 sections, 53 theorems, 407 equations.

Table of Contents

  1. Introduction and results
  2. Framework
  3. Geometry
  4. Assumption on the symbol.
  5. Symplectic sub-manifold $\Sigma$.
  6. Symplectic sub-manifold $\Sigma'$.
  7. Description and topology of $T^*_b\Omega$.
  8. Bicharacteristic and generalized flow.
  9. Statement of Theorems
  10. Semiclassical formulation
  11. Notations and pseudo-differential calculus
  12. Evolution equation and resolvent estimate
  13. A priori estimate on traces
  14. Semiclassical measure and the characteristic set
  15. Support of the semiclassical measure
  16. ...and 30 more sections

Key Result

Theorem 1

We assume that $P$, $a(x)$ and $\Omega$ satisfy the modified Geometric Control Condition given in Definition def: mGCC. We assume that $P$ has the form given in def: forme de P and $P$ is self-adjoint positive definite. Let $A$ be defined by def: forme de A, we have

Theorems & Definitions (115)

  • Definition 1.1: Finite contact with the boundary
  • Remark 1
  • Definition 1.2
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Theorem 2
  • Lemma 2.1
  • proof
  • ...and 105 more