Wave equation with Robin condition, quantitative estimates of strong unique continuation at the boundary

TL;DR

This paper provides a sharp quantitative estimate of strong unique continuation at the boundary for solutions to the wave equation with Robin boundary conditions, extending understanding of boundary behavior and vanishing properties.

Contribution

It introduces a novel quantitative estimate for boundary unique continuation of wave equations with Robin conditions, complementing the known qualitative properties.

Findings

Established a sharp quantitative boundary unique continuation estimate.

Proved that solutions with flat boundary data vanish locally.

Extended the understanding of boundary behavior for wave equations.

Abstract

The main result of the present paper consists in a quantitative estimate of unique continuation at the boundary for solutions to the wave equation. Such estimate is the sharp quantitative counterpart of the following strong unique continuation property: let $u$ be a solution to the wave equation that satisfies an homogeneous Robin condition on a portion $S$ of the boundary and the restriction of $u_{\mid S}$ on $S$ is flat on a segment $\{0\}\times J$ with $0\in S$ then $u_{\mid S}$ vanishes in a neighborhood of $\{0\}\times J$.