Carleman estimates and unique continuation for solutions to boundary value problems
Let \(K\) be a domain in \(\mathbb{R}^n\) and \(S\) a smooth surface that intersects \(\partial K\) and divides \(K\) into two parts. The paper is devoted to establishing Carleman-type inequalities for solutions \(u\) of very general boundary value problems. These inequalities imply that (i) \(u\) can be uniquely continued across \(S\) and (ii) the regularity properties of \(u\) ``extend across \(S\)''. The hypotheses involved are, apart from a relaxed pseudoconvexity condition on \(S\), what the author calls a strong or an \(r\)-strong Lopatinskiĭ condition for the boundary operator with respect to a point of \(S\cap\partial K\). The paper is necessarily quite technical, but the assumptions are illustrated for the wave, Schrödinger, and plate equations.
scientific article; zbMATH DE number 917931
scientific article