On a singular semilinear elliptic boundary value problem and the boundary Harnack principle

1999/06/09 by Siva Athreya, Athreya, Siva
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.math/9906213

arxiv created 1999/06/09 · openalex publication_date 1999/06/09 · openalex created_date 2016/06/24 · arxiv updated 2016/09/07 · openalex updated_date 2026/07/28

Abstract

We consider the singular boundary-value problem Δu = f(u) in D; u|_dD= phi, where 1. D is a bounded C²-domain of R^d, d >= 3 2. f: (0,1) -> (0,1) is a locally Hölder continuous function such that f(u) -> 1 as u -> 0 at the rate u^-α, for some αin (0,1), 3. and phi is a positive continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in D-bar. Such solutions are shown to satisfy a boundary Harnack principle. Probabilistic techniques are used in proving the main results.

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