Wiener regularity and heat semigroups on spaces of continuous functions. (English) Zbl 0920.35041

Escher, Joachim (ed.) et al., Topics in nonlinear analysis. The Herbert Amann anniversary volume. Basel: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 35, 29-49 (1999).
The authors examine the Dirichlet Laplacian \(\Delta_0\) on the Banach space \((C_0(\Omega), \|\cdot\|_\infty)\) where \(\Omega\) is an open subset of \(\mathbb{R}^N\) and \[ C_0(\Omega)= \Biggl\{f: \Omega\to \mathbb{C}\text{ continuous}:\;\lim_{x\to z} f(x)= 0\;\forall z\in\partial\Omega,\quad \lim_{\substack{| x|\to \infty\\ x\in\Omega}} f(x)= 0\Biggr\}. \] The Laplace operator \(\Delta_0\) is defined on \(C_0(\Omega)\) with maximal distributional domain; i.e. \[ D(\Delta_0)= \{f\in C_0(\Omega): \Delta f\in C_0(\Omega)\},\quad \Delta_0f= \Delta f, \] where \(\Delta f\) denotes the distributional Laplacian of \(f\). It is shown that \(\Delta_0\) generates a holomorphic \(C_0\)-semigroup on \(C_0(\Omega)\) if and only if \(\Omega\) is regular in the sense of Wiener. A corresponding result is also given for more general elliptic operators for the case where \(\Omega\) is bounded.
For the entire collection see [Zbl 0903.00112].
Reviewer: W.Lamb (Glasgow)


35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47D06 One-parameter semigroups and linear evolution equations
35B65 Smoothness and regularity of solutions to PDEs