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

Coscia, Alessandra; Mingione, Giuseppe

C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 4, 363-368 (1999).

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