Existence and regularity of extremal solutions for a mean-curvature equation

2009/04/03 by Antoine Mellet, Mellet, Antoine, Julien Vovelle +1
Computer Science · Mathematics · #53A10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #msc:53A10

paper · pdf · doi:10.48550/arxiv.0904.0618

v2: Typos corrected. Proof of Theorem 2.11 added.

openalex publication_date 2009/04/03 · arxiv created 2010/04/14 · arxiv updated 2010/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a class of mean curvature equations -\mathcal Mu=H+λu^p where \mathcal M denotes the mean curvature operator and for p≥ 1. We show that there exists an extremal parameter λ^* such that this equation admits a minimal weak solutions for all λ∈ [0,λ^*], while no weak solutions exists for λ>λ^* (weak solutions will be defined as critical points of a suitable functional). In the radially symmetric case, we then show that minimal weak solutions are classical solutions for all λ∈ [0,λ^*] and that another branch of classical solutions exists in a neighborhood (λ_*-η,λ^*) of λ^*.

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