Fundamental solutions for anisotropic elliptic equations: existence and a priori estimates

Florica C. Cîrstea, Jérôme Vétois

Abstract

We study anisotropic elliptic equations with the right-hand side the Dirac mass at zero. We introduce a suitable notion of fundamental solution (or Green's function) and establish its existence, together with sharp pointwise upper bound estimates near the origin for the solution and its derivatives. The latter is based on a Moser-type iteration scheme specific to each case, which is intricate due to our anisotropic analogue of the reverse Hölder inequality. We also derive some generalized anisotropic Sobolev inequalities and estimates in weak Lebesgue spaces as critical tools in our proof.

Keywords: Anisotropic equations, Green's function, Moser-type iteration scheme.