Some remarks on the higher regularity of minimizers of anisotropic functionals

Siepe, Francesco

doi:10.21608/ejmaa.2024.252405.1100

Some remarks on the higher regularity of minimizers of anisotropic functionals

Document Type : Regular research papers

Author

Francesco Siepe

University of Salerno

10.21608/ejmaa.2024.252405.1100

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 $\Omega$.

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Main Subjects