Morrey estimates for the gradient in non-linear variational transmission problems

TL;DR

This paper proves local Morrey-space regularity for the gradient of minimizers in nonlinear variational transmission problems with discontinuous coefficients, extending regularity results to a class of problems with $p$-growth conditions.

Contribution

It establishes Morrey estimates for the gradient in a nonlinear transmission setting with discontinuous coefficients, a novel regularity result in this context.

Findings

Gradient belongs to local Morrey space $L^{2,\lambda}_{ ext{loc}}$ for $0 extless\lambda extless n$.

Regularity holds for $p$-growth nonlinear energies with $2 extless p extless rac{2n}{n-2}$.

Quantitative decay estimates are used to extend flat interface results to general interfaces.

Abstract

We study a class of variational transmission problems driven by nonlinear energies with discontinuous coefficients across a prescribed interface. The model setting consists of integral functionals of the form \[ \mathcal{F}(u;E)=\int_{\Omega}\sigma_E(x)\,F(\nabla u)\,dx, \] where the coefficient $\sigma_E$ takes two constant values on complementary regions separated by a $C^1$ hypersurface, and the integrand $F$ satisfies standard $p$-growth and monotonicity conditions with $p>2$. In this nonlinear variational framework, we establish local Morrey-space regularity for the gradient of local minimizers, proving that $\nabla u\in L^{2,\lambda}_{\mathrm{loc}}(\Omega)$ for every $0\leq\lambda<n$, provided $2<p<\frac{2n}{n-2}$. The proof is based on quantitative decay estimates for the energy near the interface, first obtained in a flat configuration and then extended to the general case by a suitable approximation argument.