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

Abstract

We study anisotropic equations such as (with Dirac mass δ0 at 0) in a domain Ω ⊂ ℝ n (n ≥ 2) with 0 ∈ Ω and u|∂Ω = 0. Suppose that p i ∈ (1, ∞) for all i with their harmonic mean p satisfying either Case 1: p < n and or Case 2: p = n and Ω 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.