Anisotropic Minimal Surfaces and degenerate PDEs

When studying the regularity of surfaces which locally minimize the functional \int \| \nu_{E}\|_{p} for p > 2, one quickly runs into the pseudo p-Laplacian: a differential equation which, in this anisotropic setting, plays a role analogous to the role that the Laplacian plays for area minimizers. When the surface is a graph over the plane orthogonal to any standard basis vector, for instance e_{n}, the observation that D^{2}|_{(\cdot, 1)} \| \cdot \|_{p} \equiv 0 causes major problems for this regularity theory. Isolating the roles of homogeneity from the usual definition of ellipticity, we can consider any strictly convex norm \rho on \mathbb{R}^{n} and \gamma\in (1,n) , and recover De Giorgi-Nash-Moser theory whenever one considers weak solutions of \int \langle \rho(Du)^{\gamma-1} (D \rho)(Du), D \varphi \rangle = 0 . Time permitting, some preliminary results on 1st-order regularity of u stemming from a power-type convexity on \rho will also be discussed.