The probabilistic solution of the third boundary value problem for second order elliptic equations
type: Item
instance of: scholarly article
title: The probabilistic solution of the third boundary value problem for second order elliptic equations
zbMATH Open document ID: 0688.60063
DOI: 10.1007/BF01217746
author: Vassilis G. Papanicolaou
publication date: 1990-01-01T00:00:00Z
review text: Using standard reflected Brownian motion (SRBM) and martingales we define (in the spirit of Stroock and Varadhan) the probabilistic solution of the boundary value problem \[ 2^{-1}\Delta u+qu=0,\quad in\quad D;\quad \partial u/\partial n+cu=-f,\quad on\quad \partial D, \] where D is a bounded domain with \(C^ 3\) boundary and n is the inward unit normal vector on \(\partial D\). The assumptions for q, c and f are quite general. The corresponding Dirichlet problem was studied by Chung, Rao, Zhao and others and the corresponding Neumann by Pei Hsu. Here we show that the probabilistic solution of our problem exists, is unique (unless we hit an eigenvalue), continuous on \(\bar D\) and equivalent to the weak analytic solution. The method we use is to reduce the problem to an integral equation in D that involves the associated semigroup and, hence, to the study of the properties of this semigroup. In this way we do not have to assume that the spectrum is negative (almost every previous work on these probabilistic solutions makes this assumption). We construct the kernel of this semigroup and we prove certain estimates for it which help us to establish many other results, including the gauge theorem. We also show that, if the boundary function c is continuous, our semigroup is a uniform limit of Neumann semigroups and, furthermore, that the Dirichlet semigroup is a uniform limit of semigroups of our type. Therefore the Dirichlet spectrum is a ``monotone'' limit of spectra of mixed problems, a fact which is mentioned without proof in vol. 1, ch. IV, sec. 2 of the Methods of Mathematical Physics by Courant and Hilbert. This establishes the interrelation of the three boundary value problems. Finally, we add a drift term to our differential equation, which becomes \[ 2^{-1}\Delta u+b\cdot \nabla u+qu=0 \] and we solve the third boundary value problem for this equation probabilistically, with the help of Girsanov's transformation.
reviewed by: Vassilis G. Papanicolaou
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60J57
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 4128197
zbMATH Keywords: Schrödinger semigroup
zbMATH Keywords: Feynman-Kac functional
zbMATH Keywords: standard reflected Brownian motion
zbMATH Keywords: boundary value problem
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Neumann semigroups
zbMATH Keywords: Dirichlet semigroup
zbMATH Keywords: Dirichlet spectrum
zbMATH Keywords: Girsanov's transformation
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4128197