Calculus of Variations and Geometric Measure Theory

G. Cupini - P. Marcellini - E. Mascolo

Local boundedness of minimizers with limit growth conditions

created by cupini on 22 Dec 2014
modified on 14 Oct 2016


Accepted Paper

Inserted: 22 dec 2014
Last Updated: 14 oct 2016

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


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 \[ c_1\sum_{i=1}^n
^{p_i} \le {f}(x,s,\xi)\le c_2\left\{1+
^{q}\right\}, \] 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