A limiting case in partial regularity for quasiconvex functionals

TL;DR

This paper investigates the partial regularity of local minimizers in quasiconvex variational problems, showing that under certain borderline conditions on the data, the gradient exhibits almost everywhere BMO-regularity.

Contribution

It establishes a new limiting case in partial regularity theory for quasiconvex functionals with data in the borderline Marcinkiewicz space.

Findings

Gradient of minimizers is almost everywhere BMO-regular under borderline conditions.

Provides a new threshold case in partial regularity for quasiconvex functionals.

Extends regularity results to nonhomogeneous integrals with specific growth conditions.

Abstract

Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $p$-growth of the type $$ w \mapsto \int \left[F(Dw)-f\cdot w\right]dx $$ feature almost everywhere $\mbox{BMO}$-regular gradient provided that $f$ belongs to the borderline Marcinkiewicz space $L(n,\infty)$.