Higher integrabilities and boundednesses for minimizers of weighted anisotropic integral functionals
Abstract
We consider the weighted anisotropic integral functional $$I(u)=\int_{\Omega}f(x,Du(x))dx,$$ where $\Omega\subset R^n$ is a bounded open set, $u:\Omega\subset R^n \rightarrow R $, $f:\Omega \times R^n \rightarrow [0,+\infty) $ is a Carath\'{e}odory function satisfying$$ \sum_{i=1}^{n}v_{i}{|z_{i}|}^{p_{i}}\leq f(x,z)\leq c\left(1+\sum_{i=1}^{n}v_{i}{|z_{i}|}^{q_{i}}\right),$$\\in which $c>0 $ is a constant, $1<p_i<q_i<n$, $i=1,2,\cdots,n$, ${\nu_i}$ is the positive weighted function on $\Omega$ and\\$${\nu _i} \in L_{loc}^1(\Omega ),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\left( {\frac{1}{{{\nu _i}}}} \right)^{{m_i}}} \in {L^1}(\Omega ),{m_i} \ge \frac{1}{{{p_i} - 1}}.$$\\By using the weighted anisotropic Sobolev inequality and the iteration Lemma, it is proven the higher integrability for the minimizer $u$ of $I(u)$ when the boundary datum has the higher integrability. We also obtain the global boundednesses of exponential form and $L^\infty(\Omega)$ for the minimizer, respectively. Furthermore, the similar results for the minimizer of the obstacle problem to $I(u)$ are given.Keywords
weighted anisotropic integral functional, minimizer, higher integrability, boundedness