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

Huy Q. Nguyen

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arXiv · 2206.02321

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

Huy Q. Nguyen

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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 $C^\alpha$, $\alpha\in (0, 1)$.

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Huy Q. Nguyen. 2022-06-06. Coercivity of the Dirichlet-to-Neumann operator and applications to the Muskat problem. https://arxiv.org/abs/2206.02321

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