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arXiv · 1707.05556

The Dirichlet-to-Neumann operator on $C(\partial \Omega)$

W. Arendt·A. F. M. ter Elst
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Abstract

Let $\Omega \subset {\bf R}^d$ be an open bounded set with Lipschitz boundary $\Gamma$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential $V$. We show that the semigroup generated by $-D_V$ leaves $C(\Gamma)$ invariant and that the restriction of this semigroup to $C(\Gamma)$ is a $C_0$-semigroup. We investigate positivity and spectral properties of this semigroup. We also present results where $V$ is allowed to be negative. Of independent interest is a new criterium for semigroups to have a continuous kernel.

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W. Arendt, A. F. M. ter Elst. 2017-07-18. The Dirichlet-to-Neumann operator on $C(\partial \Omega)$. https://arxiv.org/abs/1707.05556

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