Accepted Paper

Inserted: 11 mar 2025
Last Updated: 11 mar 2025

Journal: Calc. Var. Partial Differ. Equ.
Year: 2025

Abstract:

We show that local minimizers of the non-autonomous functional \[ \mathcal P_{\log}(u,\Omega)= \int_\Omega
Du
^p\big(1+a(x)\log (e+
Du
)\big)\,dx,\qquad p>1, \] have continuous gradient provided that the function $a(\cdot)$ is (almost everywhere) non-negative and weakly differentiable, and moreover its gradient locally belongs to the Lorentz-Zygmund space $L^{n,1}\log L$. This gives a precise insight of the fact that for this type of two-phase functionals the lack of uniform ellipticity can be overcome by additional regularity of the switching coefficient $a(\cdot)$; the novelty is that the condition is not pointwise, but has integral character, and actually improves the known results ensuring regularity for minimizers of such functionals.

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