Fundamental Solutions for Anisotropic Elliptic Equations: Existence and A Priori Estimates

We study anisotropic equations such as (with Dirac mass delta(0) at 0) in a domain omega subset of Double-struck capital R- n (n >= 2) with 0 is an element of omega and u|( partial differential omega) = 0. Suppose that p ( i ) is an element of (1, infinity) for all i with their harmonic mean p satisfying either Case 1: p < n and or Case 2: p = n and omega is bounded. We establish the existence of a suitable notion of fundamental solution (or Green's function), together with sharp pointwise upper bound estimates near zero via an anisotropic Moser-type iteration scheme. As critical tools, we derive generalized anisotropic Sobolev inequalities and estimates in weak Lebesgue spaces.