Accepted Paper

Inserted: 31 jul 2026
Last Updated: 31 jul 2026

Journal: Discrete and Continuous Dynamical Systems - Series S
Volume: 12
Number: (2)
Pages: 251-265
Year: 2019
Doi: 10.3934/dcdss.2019018

Abstract:

We are interested in the \textit{regularity} of local minimizers of energy integrals of the Calculus of Variations. Precisely, let $\Omega $ be an open subset of $\mathbb{R}^{n}$. Let $f\left( x,\xi \right) $ be a real function defined in $\Omega \times \mathbb{R}^{n}$ \ satisfying the growth condition $% \left\vert f_{\xi x}\left( x,\xi \right) \right\vert \leq h\left( x\right) \left\vert \xi \right\vert ^{p-1}$, for% $x\in \Omega $ and $\xi \in \mathbb{R}^{n}$ with $\left\vert \xi \right\vert \geq M_{0}$ for some $M_{0}\geq 0$, with $h\in L_{{\rm loc}}^{r}\left( \Omega \right) $ for some $r>n$. This growth condition is more general than those considered in the mathematical literature and allows us to handle some cases recently studied in similar contexts. We associate to $f\left( x,\xi \right) $ the so-called \textit{natural }$p-$\textit{growth conditions} on the second derivatives $f_{\xi \xi}\left( x,\xi \right) $; i.e., $% \left( p-2\right) -$growth for $\left\vert f_{\xi \xi}\left( x,\xi \right) \right\vert $ from above and $\left( p-2\right) -$growth from below for the quadratic form $(f_{\xi \xi}\left( x,\xi \right) \lambda, \lambda)$; for details see either (1.3) or (2.2) below. We prove that these conditions are sufficient for the \textit{local Lipschitz continuity} of any minimizer $% u\in W_{{\rm loc}}^{1,p}\left( \Omega \right) $ of the energy integral $% \int_{\Omega }f\left( x,Du\left( x\right) \right) \,dx\,$.

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