## 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

Zbl 0812.35042
Full Text:

### References:

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