A Harnack inequality for weak solutions of anisotropic PDEs

We'll introduce a broad class of PDEs which arise from the Calculus of Variations. After producing specific examples of some PDEs that fall within this class and stating new results about the regularity of solutions we outline a Moser Iteration based argument to derive a Harnack inequality for weak solutions. This demonstrates that for 0th order regularity, the aspect of "ellipticity" which is useful is the fixed homogeneity. This raises the question of whether or not some notion of convexity can be used to replace ellipticity and still recover a robust theory for 1st order regularity of solutions to anisotropic PDEs.