Calculus of Variations and Geometric Measure Theory

G. P. Leonardi - G. Saracco

The prescribed mean curvature equation in weakly regular domains

created by leonardi on 15 Jun 2016
modified by saracco on 15 Feb 2020


Published Paper

Inserted: 15 jun 2016
Last Updated: 15 feb 2020

Journal: NoDEA Nonlinear Differential Equations Appl.
Volume: 25
Number: 2
Pages: 9
Year: 2018
Doi: 10.1007/s00030-018-0500-3

ArXiv: 1606.04828 PDF

The results on the weak normal trace of vector fields have been now extended and moved in a self-contained paper available at:


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 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 weak Young's law for $(\Lambda,r_{0})$-minimizers of the perimeter.

Keywords: perimeter, capillarity, prescribed mean curvature, weal normal trace