Accepted Paper

Inserted: 22 dec 2014
Last Updated: 14 oct 2016

Journal: J. Optim. Theory Appl.
Year: 2015

Abstract:

The energy-integral of the calculus of variations \[\mathcal{F}(u;\Omega):=\int_{\Omega }f(x,u,Du(x))\,dx, \] where $\Omega$ is an open bounded subset of $\mathbb{R}^n$, $n\ge 2$, and $f: \Omega\times \mathbb{R}\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ is a Carath\'{e}odory function, $f(x,s,\xi)$ convex in $(s,\xi)$ and satisfying the growth condition \[ c_1\sum_{i=1}^n
\xi_i
^{p_i} \le {f}(x,s,\xi)\le c_2\left\{1+
s
^q+
\xi
^{q}\right\}, \] has a limit behavior when $q=n\overline{p}/(n-\overline{p})$, where $\overline{p}$ is the harmonic average of the exponents $p_i$, $i=1,\ldots,n$. In fact, if $q$ is larger than in the stated equality, counterexamples to the local boundedness and regularity of minimizers are known. In this paper we prove the local boundedness of minimizers (and also of quasi-minimizers) under this stated limit condition. Some other general and limit growth conditions are also considered.

Keywords: local boundedness, quasi-minimizer, anisotropic growth condition