Boundary stabilization of transmission problems

TL;DR

This paper investigates boundary stabilization in transmission problems with dissipative boundaries, establishing resolvent bounds, eigenvalue-free regions, and exponential energy decay for solutions.

Contribution

It provides new uniform resolvent bounds and eigenvalue-free regions for transmission problems with dissipative boundary conditions, leading to energy decay results.

Findings

Uniform resolvent bounds at high frequency

Eigenvalue-free regions established

Exponential decay of solution energy

Abstract

We study the transmission problem in bounded domains with dissipative boundary conditions. Under some natural assumptions, we prove uniform bounds of the corresponding resolvents on the real axis at high frequency, and as a consequence, we obtain free of eigenvalues regions. As an application, we get exponential decay of the energy of the solutions of the correpsonding mixed boundary value problems.