Sharp regularity for general Poisson equations with borderline sources

This article concerns optimal estimates for non-homogeneous degenerate elliptic equation with source functions in borderline spaces of integrability. We deliver sharp Hölder continuity estimates for solutions to $p$-degenerate elliptic equations in rough media with sources in the weak Lebesgue space $L_\text{weak}^{\frac{n}{p} + ε}$. For the borderline case, $f \in L_\text{weak}^{\frac{n}{p}}$, solutions may not be bounded; nevertheless we show that solutions have bounded mean oscillation, in particular John-Nirenberg's exponential integrability estimates can be employed. All the results presented in this paper are optimal. Our approach is based on powerful Caffarelli-type compactness methods and it can be employed in a number order situations.

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