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Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions. (English) Zbl 0727.49021

In the paper, functionals of the form \(F(u,\Omega)=\int_{\Omega}f(| Du|)dx\) are considered, where \(\Omega\) is a bounded open subset of \({\mathbb{R}}^ n\), and u varies among all functions in \(W^{1,1}_{loc}(\Omega;{\mathbb{R}}^ m)\). A function \(u\in W^{1,1}_{loc}(\Omega;{\mathbb{R}}^ m)\) is called a local minimizer for F if for every function \(w\in W^{1,1}(\Omega;{\mathbb{R}}^ m)\) with compact support F(u,spt w)\(\leq F(u+w,spt w)\). The main result of the paper is the following higher integrability result.
Assume that f: \([0,+\infty [\to [0,+\infty [\) is convex, increasing and that there exist \(1<p\leq q\) such that \(f(t)/t^ p\) is increasing, \(f(t)/t^ q\) is decreasing, and \(p>nq/(n+q)\). Then, if u is a local minimizer for F, there exists \(r>1\) such that \(f(| Du|)\in L^ r_{loc}(\Omega).\)
The same result holds for functionals of the form \(\int_{\Omega}g(x,u,Du)dx\) provided there exists a function f with the properties above such that f(\(| z| (\leq g(x,s,z)\leq xc(1+f(| z|))\forall x,s,z\).
Reviewer: G.Buttazzo (Pisa)

MSC:

49N60 Regularity of solutions in optimal control
49J40 Variational inequalities
Full Text: DOI

References:

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