The role of absorption terms in Dirichlet problems for the prescribed mean curvature equation

2023/07/26 by Oliva, Francescantonio, Petitta, Francesco, de León, Sergio Segura
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.14154

Abstract

In this paper we study existence and uniqueness of solutions to Dirichlet problems as \begincases g(u) -\rm div((D u)/(√(1+|D u|²))) = f amp; in Ω,
\newline u=0 amp; on ∂Ω, \endcases where Ω is an open bounded subset of ℝ^N (N≥ 2) with Lipschitz boundary, g:ℝ→ℝ is a continuous function and f belongs to some Lebesgue spaces. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term g(u) in order to get a solutions for data f merely belonging to L¹(Ω) and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in L^N(Ω) as well as uniqueness if g is increasing.

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