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Resolvent estimates and local energy decay for hyperbolic equations. (English) Zbl 1142.35059
The authors study the resolvent of the Laplacian in an exterior domain in \(\mathbb R^n, n \geq 2\), with \(C^\infty\) boundary and Dirichlet boundary condition. They consider the cut-off resolvent
\[ R_\chi(\lambda)= \chi (-\Delta_D-\lambda)^{-1} \chi, \] with \(\chi \in C^\infty_0(\mathbb R^n)\) equal to one in a neighborhood of the obstacle. They assume that \(R_\chi(\lambda)\) has no poles for \( \operatorname{Im} \lambda \geq - \delta\), \(\delta >0\), i.e. that there are no resonances in this domain, and they prove the estimate
\[ \left| R_\chi (\lambda) \right| \l_{L^2 \rightarrow L^2}\leq C | \lambda| ^{n-2}, \quad \lambda \in\mathbb R,\;| \lambda| \geq C_0. \] This estimate is used to prove the decay of local energies, and, moreover, the spectrum of the Lax-Phillips semigroup is studied for trapping obstacles having at least one trapped ray.

MSC:
35P25 Scattering theory for PDEs
35P15 Estimates of eigenvalues in context of PDEs
35L05 Wave equation
47A40 Scattering theory of linear operators
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