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

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 $$\mathcal{F}(u;E)={\int }_{\mathrm{\Omega }}{\sigma }_E(x)F(\mathrm{\nabla }u)dx,$$ 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 $\mathrm{\nabla }u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,\lambda }(\mathrm{\Omega })$ for every $0\le \lambda <n$, 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.