Degenerate Elliptic Partial Differential Equations in Rough Geometries

Abstract

In this thesis, we investigate the existence of weak solutions to degenerate, linear, elliptic second-order partial differential equations in divergence form with rough coefficients whose regularity is controlled by the optimal gain found in several types of Sobolev inequalities: power gain, logarithmic gain, and no gain. This gain is determined by the roughness of the geometry associated to the vector fields defined by the coefficient matrix Q of the principal part of the equation. In the case of power gain, low order coefficients must minimally belong to certain classical weighted Lebesgue spaces. In the case of logarithmic gain, they must be exponentially integrable. In the case of no gain, low order coefficients must be bounded. This investigation is conducted using the techniques of Lebesgue spaces, Orlicz spaces, and general functional analysis.