Published Paper

Inserted: 11 mar 2025
Last Updated: 11 mar 2025

Journal: Electronic Journal of Differential Equations
Volume: 2022
Number: 80
Year: 2022

Abstract:

We prove $C^1$ regularity for local vectorial minimizers of the non-autonomous functional \[ w\in W^{1,1}_{\rm loc}(\Omega;\mathbb R^N)\longmapsto \int_{\Omega}b(x)\big[
Dw
^p+a(x)
Dw
^p\log(e+
Dw
)\big] \,dx\,, \] with $\Omega$ open subset of $\mathbb R^n$, $n\geq2$ , $p>1$, $0\leq a(\cdot)\leq \
a\
_{L^{\infty}(\Omega)}<\infty$ and $0<\nu\leq b(\cdot)\leq L$. The result is obtained provided that the function $a(\cdot)$ is $\log$-Dini continuous and that the coefficient $b(\cdot)$ is Dini continuous or it is weakly differentiable and its gradient locally belongs to the Lorentz space $L^{n,1}(\Omega;\mathbb R^n)$.

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