Coercivity of the Dirichlet-to-Neumann operator and applications to the Muskat problem

Abstract

We consider the Dirichlet-to-Neumann operator in strip-like and half-space domains with Lipschitz boundary. It is shown that the quadratic form generated by the Dirichlet-to-Neumann operator controls some sharp homogeneous fractional Sobolev norm. As an application, we prove that the global Lipschitz solutions constructed in \cite{DGN} for the one-phase Muskat problem decays exponentially in time in any Hölder norm $C^{\alpha }$, $\alpha \in (0,1)$.

arXiv commentTo appear in Acta Mathematica Vietnamica, a special issue in honor of Professor Duong Minh Duc