×

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

Citations:

Zbl 0961.47020
PDF BibTeX XML Cite
Full Text: DOI