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 Hölder one, when they fail at any finite order (as proved in [9]) 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 Hölder, 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.
ToolsThe present section is devoted to present the tools and preliminary results we will need in order to prove our statements.Hence, in a first time, we will introduce the Littlewood-Paley decomposition and the definition of Besov spaces that will be used in the next subsection, in order to analyse the classes of coefficients we deal with, and to state some of their properties. There, we will also focus on Zygmund type conditions in more detail. In particular, we will compare the "integral Zygmund" condition and the BV classes.