Some remarks on the higher regularity of minimizers of anisotropic functionals | ||
| Electronic Journal of Mathematical Analysis and Applications | ||
| Volume 12, Issue 2, 2024, Pages 1-11 PDF (244.66 K) | ||
| Document Type: Regular research papers | ||
| DOI: 10.21608/ejmaa.2024.252405.1100 | ||
| Author | ||
| Francesco Siepe* | ||
| University of Salerno | ||
| Abstract | ||
| We consider the anisotropic integral functional of the calculus of variations $$ \int_{\Omega} \left[\sum_{i=1}^n c_i |D_iu|^{p_i} \right] dx, $$ where $\Omega\subset {\mathbb R}^n (n\ge 2),$ is an open bounded set, $u:\Omega\to\mathbb R$, $c_i\ge 0$ for $i=1,\dots, n$ are constants and $f_i:\mathbb R \to \mathbb [0,+\infty)$ are functions satisfying, for every $t\in \mathbb R$ the following non standard growth condition: $$ \lambda |t|^{p_i} \le f_i(t) \le \Lambda |t|^{p_i} $$ for every $i=1,\dots, n$ and some positive constants $0<\lambda<\Lambda$. Moreover, some further assumption, such as a {\it strong ellipticity} condition of the kind $$ \sum_{i=1}^n f''_i(\xi_i) \eta_i\eta_i \ge \nu \sum_{i=1}^n |\xi_i|^{p_i-2} |\eta_i|^2, $$ is assumed for every $\xi,\eta\in\mathbb R^n$ and some $\nu >0$. The aim of the article is to exhibit a minimizer of such functional, for an opportune choice of the exponents $p_i$, which turns out to be bounded everywhere and Lipschitz continuous (or even of class $C^1$) in a opportune subset of $\Omega$. | ||
| Keywords | ||
| Calculus of variations; anisotropic functional; minimizers; regularity | ||
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