$\mathrm{BMO}$ $\varepsilon$-regularity results for solutions to Legendre-Hadamard elliptic systems

TL;DR

This paper proves an epsilon-regularity result for weak solutions to Legendre-Hadamard elliptic systems, showing that small BMO norm of the gradient implies regularity, with extensions to boundary and more general systems.

Contribution

It establishes a new epsilon-regularity criterion for elliptic systems based on BMO smallness, including boundary regularity and extensions to quasilinear and higher-order systems.

Findings

Regularity results up to the boundary.

Global regularity consequences.

Extensions to quasilinear and higher-order systems.

Abstract

We will establish an $\varepsilon$-regularity result for weak solutions to Legendre-Hadamard elliptic systems, under the a-priori assumption that the gradient $\nabla u$ is small in $\mathrm{BMO}.$ Focusing on the case of Euler-Lagrange systems to simplify the exposition, regularity results will be obtained up to the boundary, and global consequences will be explored. Extensions to general quasilinear elliptic systems and higher-order integrands is also discussed.