Off-diagonal bounds for the Dirichlet-to-Neumann operator

Abstract

Let $\Omega$ be a bounded domain of ${\mathbb{R}}^{n+1}$ with $n\ge 1$. We assume that the boundary $\Gamma$ of $\Omega$ is Lipschitz. Consider the Dirichlet-to-Neumann operator $N_0$ associated with a system in divergence form of size $m$ with real symmetric and H\''older continuous coefficients. We prove $L^p(\Gamma )\to L^q(\Gamma )$ off-diagonal bounds of the form$$\Vert 1_F{e}^{-t{N}_0}{1}_{E}f{\Vert }_q\lesssim (t\wedge 1{)}^{\frac{n}{q}-\frac{n}{p}}{\left(1+\frac{dist(E,F)}{t}\right)}^{-1}\Vert 1_{E}f{\Vert }_p$$for all measurable subsets $E$ and $F$ of $\Gamma$. If $\Gamma$ is $C^{1+\kappa }$ for some $\kappa >0$ and $m=1$, we obtain a sharp estimate in the sense that ${\left(1+\frac{dist(E,F)}{t}\right)}^{-1}$ can be replaced by${\left(1+\frac{dist(E,F)}{t}\right)}^{-(1+\frac{n}{p}-\frac{n}{q})}$. Such bounds are also valid for complex time. For $n=1$, we apply our off-diagonal bounds to prove that the Dirichlet-to-Neumann operator associated with a system generates an analytic semigroup on $L^p(\Gamma )$ for all $p\in (1,\infty )$. In addition, the corresponding evolution problem has $L^q(L^p)$-maximal regularity.

final version, to appear in J. Math. Anal. Appl

References in corpus (1)

• Sectorial forms and degenerate differential operators

Cited by in corpus (1)

• Weighted non-autonomous $L^q(L^p)$ maximal regularity for complex systems