Global boundedness, interior gradient estimates, and boundary regularity for the mean curvature equation with boundary conditions

Fei-Tsen Liang

doi:10.1155/s0161171204307039

Abstract

We obtain global estimates for the modulus, interior gradient estimates, and boundary Hölder continuity estimates for solutionsuto the capillarity problem and to the Dirichlet problem for the mean curvature equation merely in terms of the mean curvature, together with the boundary contact angle in the capillarity problem and the boundary values in the Dirichlet problem.

Highlights

Let Ω be a bounded domain in Rn, n ≥ 2
A solution of the Dirichlet problem can be regarded as a solution of (1.1) subject to the Dirichlet boundary condition u = φ, (1.3)
Where φ is a given function on ∂Ω; a solution of the capillarity problem can be regarded as a solution of (1.1) subject to the “contact angle” boundary condition

Summary

Introduction

Let Ω be a bounded domain in Rn, n ≥ 2. Consider a solution to the mean curvature equation div T u = H x, u(x) in Ω,. A solution of the Dirichlet problem can be regarded as a solution of (1.1) subject to the Dirichlet boundary condition u = φ,. Where φ is a given function on ∂Ω; a solution of the capillarity problem can be regarded as a solution of (1.1) subject to the “contact angle” boundary condition. Where ν is the outward pointing unit normal of ∂Ω, and where cos θ is a given function on ∂Ω. (in the capillarity problem, we are considering geometrically a function u in Ωwhose graph has the prescribed mean curvature H and which meets the boundary cylinder in the prescribed angle θ.) Here, H = H(x, t) is assumed to be a given locally Lipschitz function in Ω × R satisfying the structural conditions.

The Dirichlet problem corresponds to the variational problem

Then we set HA