Borderline global regularity for nonuniformly elliptic systems

TL;DR

This paper proves optimal global regularity, including Lipschitz continuity, for solutions to nonuniformly elliptic systems under borderline Lorentz conditions, resolving a previously open question.

Contribution

It establishes sharp global regularity results for nonuniformly elliptic systems with minimal assumptions, settling an open problem about optimal regularity conditions.

Findings

Solutions are Lipschitz continuous under borderline Lorentz assumptions.

The results are sharp and optimal, confirming the necessity of the conditions.

The paper settles the open regularity question posed in prior work.

Abstract

We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in \cite{bdms}.