Regularity results for a class of functionals with non-standard growth
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 \textit{T. Iwaniec} and \textit{A. Verde} [J. Funct. Anal. 169, No. 2, 391-420 (1999; Zbl 0961.47020)].
scientific article; zbMATH DE number 1586360