Anisotropic minimal surface equation with Dirichlet boundary condition

Abstract

This paper investigates the Dirichlet problem for the anisotropic minimal surface equation in a bounded domain. Under the natural assumption of non-negative boundary anisotropic mean curvature, we establish the unique solvability of the Dirichlet problem for continuous boundary data. To achieve this, an essential a priori gradient estimate is established, which also allows us to prove a weak version of Bernstein's theorem for entire solutions under a sharp, one-sided linear growth assumption. Moreover, using the direct method in the calculus of variations, we prove the existence and local Lipschitz regularity of generalized minimizers in $BV(\hat{I}\text{c◯})$ with $L^1(\partial \hat{I}\text{c◯})$ boundary data. We also find that this variational formulation naturally yields a Neumann-type boundary condition, geometrically explaining why the Neumann problem requires no boundary curvature constraints.

In the proof of Theorem 1.1, we have added a more detailed estimate for the term ψ'F_{ξ_iξ_j}(ν)d_{ij}. Furthermore, the choice of the test function in Lemma 4.1 is modified, yielding a precise domain of integration