## Hölder continuity of the gradient of $$p(x)$$-harmonic mappings.(English. Abridged French version)Zbl 0920.49020

The paper deals with the regularity problem for local minima of functionals of the form $F(u)= \int_\Omega| Du|^{p(x)}dx,$ where a function $$u$$ is called a local minimum for $$F$$ if $F(u)\leq F(u+ \phi)\qquad \forall\phi\in W^{1,1}_0(\Omega; \mathbb{R}^N).$ It must be noticed that the competing functions are here with vector values. The main result is that if the exponent function $$p(x)$$ is locally Hölder continuous, then every local minimizer of the functional $$F$$ has locally Hölder continuous derivatives.
A short but essential summary of the regularity results already available in the literature, together with the related references, complete the paper.
Reviewer: G.Buttazzo (Pisa)

### MSC:

 49N60 Regularity of solutions in optimal control 35B65 Smoothness and regularity of solutions to PDEs

### Keywords:

elliptic systems; Hölder continuity; regularity
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