Acta Mathematica Vietnamica
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Coercivity of the Dirichlet-to-Neumann Operator and Applications to the Muskat Problem
Huy Q. Nguyenpp. 51-62
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 norms. As an application, we prove that the global Lipschitz solutions constructed in Dong et al. (2021) for the one-phase Muskat problem decays exponentially in time in any Hölder norm $C^\alpha, \alpha \in (0,1)$.