We consider the Dirichlet problem for so-called G-minimal graphs in two dimensions. These are immersions of minimal surface type which can be presented as graphs over a planar domain. With the aid of a weight matrix G we derive a quasilinear elliptic and homogeneous differential equation for this height function. Then we solve the Dirichlet problem over convex domains Ω without differentiability assumptions and continuous boundary data with a constructive continuity and approximation method.
We firstly establish an a priori C1+α-estimate up to the boundary of the solution as we take the dense problem class of strictly convex C2+α-domains and C2+α-boundary data. By proving theorems on stability and compactness of graphs we solve this boundary value problem with a nonlinear continuity method. Then we introduce weighted conformal parameters in the graph and consider the parametric problem on the unit disc. Finally, we solve the original Dirichlet problem by using an approximation argument and the important parametric compactness theorem.
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