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

Abstract

We will establish an $\epsilon$-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.

37 pages, 1 figure. Accepted version