## Regularity results for a class of functionals with non-standard growth.(English)Zbl 0984.49020

The very interesting paper under review deals with regularity properties of local minimizers for the functional $F(u,\Omega)=\int_\Omega f(x,Du) dx$ with $$f(x,z)$$ being a Carathéodory integrand which satisfies the non-standard growth condition $|z|^{p(x)}\leq f(x,z)\leq L(1+|z|^{p(x)})\qquad \forall z\in{\mathbb R}^n,\quad x\in \Omega,$ where $$p\colon\;\Omega\to(1,+\infty)$$ is a continuous function with modulus of continuity $$\omega(R).$$
Assuming convexity of $$F$$ and $\limsup_{R\to 0} \omega(R)\log\left({1}\over{R}\right)=0,$ the authors derive $$C^{0,\alpha}_{\text{ loc}}$$-smoothness $$\forall \alpha\in(0,1),$$ of any $$W^{1,1}_{\text{ loc}}$$ minimizer of $$F(u,\Omega).$$ Further, $$C^2$$-regularity of $$f(x,z)$$ with respect to $$z$$ and $$p\in C^{0,\alpha},$$ $$\alpha\in(0,1],$$ imply local Hölder continuity of $$Du.$$ The technique used relies on perturbation at the growth exponent $$p(x)$$ and precise estimates in the Orlicz space $$L\log L(\Omega)$$ due to T. Iwaniec and A. Verde [J. Funct. Anal. 169, No. 2, 391-420 (1999; Zbl 0961.47020)].

### MSC:

 49N60 Regularity of solutions in optimal control 35B65 Smoothness and regularity of solutions to PDEs 49J45 Methods involving semicontinuity and convergence; relaxation

Zbl 0961.47020
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