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

TL;DR

This paper investigates the coercivity properties of the Dirichlet-to-Neumann operator in Lipschitz domains and applies these results to demonstrate exponential decay of solutions in the Muskat problem.

Contribution

It establishes the coercivity of the Dirichlet-to-Neumann operator in Lipschitz domains and applies this to prove decay rates for solutions of the Muskat problem.

Findings

Quadratic form controls sharp fractional Sobolev norm

Global Lipschitz solutions decay exponentially in Hölder norms

Results applicable to strip-like and half-space domains

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 H\"older norm $C^\alpha$, $\alpha\in (0, 1)$.