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Hölder continuity of minimizers of functionals with variable growth exponent. (English) Zbl 0878.49010

The paper deals with the Hölder continuity of local minimizers of integral functionals whose model is \[ F_0(u)= \int_\Omega|Du|^{a(x)}dx. \] Here \(\Omega\) is an open subset of \(\mathbb{R}^n\), the functions \(u\) are scalar, and the function \(a(x)\) is assumed to be in \(W^{1,s}(\Omega)\) with \(s>n\) and such that for a suitable \(p_0\in]1,n]\), \[ p_0\leq a(x)\leq p^*_0\quad\text{on }\Omega.\tag{1} \] The main result is that under these assumptions every local minimizer of \(F_0\) is Hölder continuous. Actually, the result proved holds for quasi minimizers of \(F_0\), and so the class of functionals for which the Hölder continuity result holds is considerably wider and includes integral functionals of the form \[ F(u)=\int_\Omega f(x,u,Du)dx, \] where the integrand \(f\) satisfies the growth assumption \[ c_1(|z|^{a(x)}-|s|^{r(x)}- 1)\leq f(x,s,z)\leq c_2(|z|^{a(x)}+|s|^{r(x)}+ 1) \] with \(a(x)\) verifying (1) and \(0\leq r(x)\leq p^*_0\) on \(\Omega\).
The reader can find a complete discussion about functionals with different growth from below and from above in the paper by P. Marcellini [J. Differ. Equations 105, No. 2, 296-333 (1993; Zbl 0812.35042)], where also a wide bibliography can be found.
Reviewer: G.Buttazzo (Pisa)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49N60 Regularity of solutions in optimal control

Citations:

Zbl 0812.35042
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References:

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