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We consider the Dirichlet-to-Neumann operator in strip-like and half-spacedomains with Lipschitz boundary. It is shown that the quadratic form generatedby the Dirichlet-to-Neumann operator controls some sharp homogeneous fractionalSobolev norm. As an application, we prove that the global Lipschitz solutionsconstructed in \cite{DGN} for the one-phase Muskat problem decays exponentiallyin time in any H\"older norm $C^\alpha$, $\alpha\in (0, 1)$.

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