Carleman estimates for the Zaremba Boundary Condition and Stabilization of Waves

TL;DR

This paper establishes Carleman estimates for the Zaremba boundary condition, leading to new stabilization and controllability results for wave and heat equations with mixed boundary conditions.

Contribution

It introduces novel Carleman estimates for Zaremba problems and applies them to achieve stabilization and controllability results for wave and heat equations.

Findings

Logarithmic decay of wave energy with Neumann feedback

Controllability results for heat equation with Zaremba boundary condition

Extension of previous stabilization results to boundary zones with Dirichlet conditions

Abstract

In this paper, we shall prove a Carleman estimate for the so-called Zaremba problem. Using some techniques of interpolation and spectral estimates, we deduce a result of stabilization for the wave equation by means of a linear Neumann feedback on the boundary. This extends previous results from the literature: indeed, our logarithmic decay result is obtained while the part where the feedback is applied contacts the boundary zone driven by an homogeneous Dirichlet condition. We also derive a controllability result for the heat equation with the Zaremba boundary condition.