Probabilistic approach to semilinear and generalized mixed boundary value problems
type: Item
instance of: scholarly article
title: Probabilistic approach to semilinear and generalized mixed boundary value problems
zbMATH Open document ID: 0769.60071
DOI: 10.1007/BF02014579
author: Renming Song
author: Zhi-Ming Ma
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 1993-05-25T00:00:00Z
review text: In the late eighties the authors have written several papers on probabilistic approach to various types of the Schrödinger equation. A survey of methods and results is given in [\textit{Z. Ma} and \textit{R. Song}, Stochastic processes, Semin., San Diego/CA (USA) 1989, Prog. Probab. 18, 135-164 (1990; Zbl 0691.60050)]. One of the main steps in their approach is the analysis of the gauge function for the corresponding problem. They succeeded to extend the method to the case of infinite gauge which corresponds to nonnegative eigenvalues of the infinitesimal generator. In this paper the authors present a probabilistic solution to the following semilinear and generalized mixed boundary value problem: \[ \begin{aligned} -(\Delta/2+\mu I_ D)u +f_ 1(u) & =\nu I_ D\quad\text{in } D, \\ u|_{I_ 1} & =g, \\ \partial u/\partial n|_{I_ 2}- 2u\mu I_{I_ 2}+2f_ 2(u) & =2\nu I_{I_ 2},\end{aligned} \] where \(D=G-B\), \(G\) a bounded \(C^ 3\) domain in \(\mathbb{R}^ d\) and \(B\) a closed subset of \(\overline G\), \(I_ 1=\partial D\cap B\), \(I_ 2=\partial D- B\), \(\mu\) and \(\nu\) signed measures on \(\tilde D=D\cup I_ 2\) belonging to the generalized Kato class, \(f_ 1\) and \(f_ 2\) continuously differentiable functions on \(\mathbb{R}\), \(g\) a bounded measurable function on \(I_ 1\), and \(\partial/\partial n\) is the outward normal derivative. The authors first solve the linear mixed boundary value problem (for the finite gauge), and based on the obtained results they give the probabilistic approach to the above semilinear problem.
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 35M10
zbMATH DE Number: 177970
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: nonnegative eigenvalues of the infinitesimal generator
zbMATH Keywords: generalized mixed boundary value problem
reviewed by: Zoran Vondraček
MaRDI profile type: Publication
cites work: Q3794583
cites work: Solving Semilinear Partial Differential Equations With Probabilistic Potential Theory
cites work: Q3855442
cites work: Q5748693
cites work: Probabilistic approach to the neumann problem
cites work: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains
description: scientific article; zbMATH DE number 177970