Annals of Probability

A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients

Tusheng Zhang

Full-text: Open access

Abstract

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of Dirichlet processes and backward stochastic differential equations play a crucial role.

Article information

Source
Ann. Probab., Volume 39, Number 4 (2011), 1502-1527.

Dates
First available in Project Euclid: 5 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1312555806

Digital Object Identifier
doi:10.1214/10-AOP591

Mathematical Reviews number (MathSciNet)
MR2857248

Zentralblatt MATH identifier
1242.60072

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 35J25: Boundary value problems for second-order elliptic equations 31C25: Dirichlet spaces

Keywords
Dirichlet processes quadratic forms Fukushima’s decomposition Dirichlet boundary value problems backward stochastic differential equations weak solutions martingale representation theorem

Citation

Zhang, Tusheng. A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients. Ann. Probab. 39 (2011), no. 4, 1502--1527. doi:10.1214/10-AOP591. https://projecteuclid.org/euclid.aop/1312555806


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