On the existence of Feller semigroups with boundary conditions. II
type: Item
instance of: scholarly article
title: On the existence of Feller semigroups with boundary conditions. II
zbMATH Open document ID: 0898.47032
author: Kazuaki Taira
published in: Journal of Functional Analysis
publication date: 1998-10-26T00:00:00Z
review text: Let \(D\) be a bounded domain of \(\mathbb{R}^n\), with \(C^\infty\) boundary \(\partial D\). A strongly continuous semigroup \(\{T_t\}_{t\geq 0}\) of operators on \(C(\overline D)\) is called a Feller semigroup on \(\overline D\) if it is nonnegative and contractive on \(C(\overline D)\), i.e., \(f\in C(\overline D)\), \(0\leq f\leq 1\Rightarrow 0\leq T_tf\leq 1\). The infinitesimal generator \({\mathbf A}\), defined by \({\mathbf A}u= \lim_{t\downarrow 0} t^{-1}(T_tu- u)\), is known to be a diffusion operator \(A\), i.e., second-order degenerate elliptic differential operator with real coefficients, \[ Au(x)= \sum^N_{i,j= 1}a^{ij}(x) {\partial^2u\over \partial x_i\partial x_j} (x)+ \sum^N_{i= 1}b^i(x) {\partial u\over\partial x_i} (x)+ c(x)u(x), \] satisfying a second-order Ventcel' boundary condition \(L\). \(L\) contains terms corresponding to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon and the viscosity phenomenon, respectively. This paper is concerned with the converse problem: construction of a Feller semigroup \(T_t\) whose infinitesimal generator is characterized by some given diffusion operator \(A\) and Ventcel' boundary condition. \textit{P. Cattiaux} [Ann. Inst. Fourier 42, No. 3, 541-624 (1992; Zbl 0780.35023)] considered the problem for the non-characteristic case that \(\sum^N_{i,j= 1}a^{ij}(x) n_in_j> 0\) for all \(x\in\partial D\), where \(n= (n_1,\dots, n_N)\) is the unit interior normal to \(\partial D\) at \(x\). This paper considers the construction problem for the characteristic case. The author constructs a Feller semigroup corresponding to a transversal Ventcel' boundary condition that one of the reflection and viscosity phenomenon occurs at each point of \(\partial D\).
reviewed by: Sen-Yen Shaw
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 764020
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: Feller semigroup
zbMATH Keywords: infinitesimal generator
zbMATH Keywords: second-order degenerate elliptic differential operator with real coefficients
zbMATH Keywords: second-order Ventcel' boundary condition
zbMATH Keywords: diffusion operator
zbMATH Keywords: transversal Ventcel' boundary condition
zbMATH Keywords: reflection
zbMATH Keywords: viscosity
MaRDI profile type: Publication
full work available at URL: jfan.1995.1044
OpenAlex ID: W4213010566
DOI: 10.1006/JFAN.1995.1044
description: scientific article; zbMATH DE number 764020