C^1,α theory for the prescribed mean curvature equation with Dirichlet data

2010/07/20 by Theodora Bourni, Bourni, Theodora
Computer Science · Mathematics · #35J93 #49Q15 #49Q20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #math.DG #msc:35J93 #msc:49Q15 #msc:49Q20

paper · pdf · doi:10.48550/arxiv.1007.3402

45 pages, no figures, to appear in Journal of Geometric Analysis

arxiv created 2010/07/20 · openalex publication_date 2010/07/20 · arxiv updated 2010/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a C^1,α manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class C^1,α.

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