A-priori gradient bound for elliptic systems under either slow or fast growth conditions

Abstract

We obtain an a-priori $W_{loc}^{1,\infty }\ ( \Omega ;\mathbb{R}^{m}\ ) -$bound for solutions in $\Omega \subset \mathbb{R}^{n} $, $n\geq 2$, to the elliptic system \begin{equation*} \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}\ ( \frac{g_{t}\ ( x,\ |Du\ | \ ) }{\ |Du\ | } u_{x_{i}}^{\alpha }\ ) =0,\;\;\;\;\;\alpha =1,2,\ldots ,m, \end{equation*} where $g\ ( x,t\ ) $, $g:\Omega \times \ [ 0,\infty \ ) \rightarrow \ [ 0,\infty \ ) $, is a Carath\'{e}odory function, convex and increasing with respect to the gradient variable $t\in \ [ 0,\infty \ ) $. We allow $x-$dependence, which turns out to be a relevant difference with respect to the autonomous case and not only a technical perturbation. Our assumptions allow us to consider both fast and slow growth. We allow fast growth even of exponential type; and slow growth, for instance of Orlicz-type with energy-integrands such as $g\ ( x,\ | Du\ |\ ) =|Du|\log (1+|Du|)$ or, when $n=2,3$, even asymptotic linear growth with energy integrands of the type \begin{equation*} g\ ( x,\ | Du\ | \ ) =\ | Du\ | -a\ ( x\ ) \sqrt{\ | Du\ | }\,. \end{equation*}