Full Regularity for Convex Integral Functionals with $p(x)$ Growth in Low Dimensions

Habermann, Jens

Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 3, pp. 521-541

Résumé

For $\Omega \subset \mathbf{R}^{n}$; $n \ge 2$, and $N \ge 1$ we consider vector valued minimizers $u \in W_{loc}^{m,p(\cdot)}(\Omega,\mathbf{R}^{N})$ of a uniformly convex integral functional of the type $$\mathcal{F} \left[ u,\Omega \right] := \int_{\Omega} f(x,D^{m}u) \, dx,$$ where $f$ is a Carathéorody function satisfying $p(x)$ growth conditions with respect to the second variable. We show that if the dimension $n$ is small enough, dependent on the structure conditions of the functional, there holds $$D^{k}u \in C_{loc}^{0,\beta}(\Omega) \,\, \text{for} \,\, k \in \{0,\cdots,m-1\},$$ for some $\beta$, also depending on the structural data, provided that the nonlinearity exponent $p$ is uniformly continuous with modulus of continuity $\omega$ satisfying $$\limsup_{\rho\downarrow 0} \omega(\rho) \log \bigg( \frac{1}{\rho} \bigg) = 0.$$