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Regularity results for a class of quasiconvex functionals with nonstandard growth. (English) Zbl 1027.49031

The authors consider the integral functional \[ I(u)= \int _{\Omega} f(x, Du(x))\,dx, \] under nonstandard growth assumptions of \((p-q)\)-type: \[ |z|^{p(x)} \leq f(x,z)\leq L (1+|z|^{p(x)}) \] for some function \( p(x) >1\), where \(f\) is a quasiconvex density energy and under sharp assumptions on \(p\) and \(f\), a new partial regularity result is proved for the minimizers of \(I\). More precisely, let \(f\in C^{2}\) satisfy ellipticity and continuity conditions and \[ |p(x)-p(y)|\leq \omega (|x-y|) \] where \(\omega\) is a nondecreasing continuous function vanishing at zero, such that \(\omega (R)\leq LR^\alpha \) for some \(0<\alpha \leq 1\) and for any \(R\leq 1\). For \(u\in W^{1,1}_{\text{loc}}(\Omega,\mathbb R^N),\) a local minimizer of the functional \(I\), there exists an open subset \(\Omega_{0}\) of \(\Omega\), such that \(|\Omega\setminus\Omega _{0}|=0\) and that \(Du\) is locally Hölder continuous in \(\Omega_{0}\).

MSC:

49N60 Regularity of solutions in optimal control
35B65 Smoothness and regularity of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
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