On Surfaces of Prescribed Weighted Mean Curvature

TL;DR

This paper generalizes the concept of surfaces with prescribed mean curvature by incorporating a weight matrix, deriving related differential equations, and solving the Dirichlet problem for such surfaces.

Contribution

It introduces a framework for surfaces of prescribed weighted mean curvature, including differential equations and boundary value problem solutions, extending previous isotropic models.

Findings

Derived a differential equation for the normal of surfaces with prescribed weighted mean curvature.

Proved a quasilinear elliptic equation for graphs of prescribed weighted mean curvature.

Established height and boundary gradient estimates and solved the Dirichlet problem.

Abstract

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.