Generalized solutions to semilinear elliptic equations with measure data

2023/10/11 by Tomasz Klimsiak, Klimsiak, Tomasz
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2310.07447

openalex publication_date 2023/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We address an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in: Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects of Nonlinear Dispersive Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.), Annals of Mathematics Studies, 163 (2007). We prove that for any bounded Borel measure μ on a smooth bounded domain D⊂\mathbb R^d and asymptotically convex non-decreasing non-negative continuous function g on \mathbb R the sequence of solutions to the semi-linear equation (P): -Δu+g(u)=ρ_n∗μ (ρ_n is a mollifier) that is subject to homogeneous Dirichlet condition, converges to the function that solves (P) with ρ_n∗μ replaced by the reduced measure μ^* (metric projection onto the space of good measures). We also provide a corresponding version of this result without non-negativity assumption on g.

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