The prescribed mean curvature equation in weakly regular domains

Abstract

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(\Lambda ,r_0)$-minimizers of the perimeter.

23 pages, 1 figure --- The results on the weak normal trace of vector fields have been now extended and moved in a self-contained paper available at: arXiv:1708.01393

References in corpus (2)

• Weighted Cheeger sets are domains of isoperimetry
• Two examples of minimal Cheeger sets in the plane

Cited by in corpus (11)

• Anzellotti's pairing theory and the Gauss--Green theorem
• Cauchy Fluxes and Gauss-Green Formulas for Divergence-Measure Fields over General Open Sets
• An extension of the pairing theory between divergence-measure fields and BV functions
• Rigidity and trace properties of divergence-measure vector fields
• The Cheeger problem in abstract measure spaces
• Two examples of minimal Cheeger sets in the plane
• Minimizers of the prescribed curvature functional in a Jordan domain with no necks
• On the $N$-Cheeger problem for component-wise increasing norms
• A Vertex-Skipping property for almost-minimizers of the relative perimeter in convex sets
• Geometric criteria for the existence of capillary surfaces in tubes
• The isoperimetric problem in $2$d domains without necks