Weak observability estimates for 1-D wave equations with rough coefficients

TL;DR

This paper establishes observability estimates for 1-D wave equations with rough coefficients, revealing how the regularity of coefficients affects the ability to estimate solutions and identifying the precise loss of derivatives involved.

Contribution

It extends classical observability results to coefficients in the Zygmund class and characterizes the loss of derivatives for less regular coefficients like log-Lipschitz.

Findings

Classical observability estimates hold for Zygmund coefficients.

Observability estimates with derivative loss are proved for log-Lipschitz coefficients.

Establishes a sharp relation between coefficient regularity and derivative loss in estimates.

Abstract

In this paper we prove observability estimates for 1-dimensional wave equations with non-Lipschitz coefficients. For coefficients in the Zygmund class we prove a "classical" observability estimate, which extends the well-known observability results in the energy space for $BV$ regularity. When the coefficients are instead log-Lipschitz or log-Zygmund, we prove observability estimates "with loss of derivatives": in order to estimate the total energy of the solutions, we need measurements on some higher order Sobolev norms at the boundary. This last result represents the intermediate step between the Lipschitz (or Zygmund) case, when observability estimates hold in the energy space, and the H\"older one, when they fail at any finite order (as proved in \cite{Castro-Z}) due to an infinite loss of derivatives. We also establish a sharp relation between the modulus of continuity of the coefficients and the loss of derivatives in the observability estimates. In particular, we will show that under any condition which is weaker than the log-Lipschitz one (not only H\"older, for instance), observability estimates fail in general, while in the intermediate instance between the Lipschitz and the log-Lipschitz ones they can hold only admitting a loss of a finite number of derivatives. This classification has an exact counterpart when considering also the second variation of the coefficients.