Journal of Convex Analysis 19 (2012), No. 1, 063--090
Copyright Heldermann Verlag 2012
Differentiabilty and Partial Hölder Continuity of Solutions of Nonlinear Elliptic Systems
Giuseppe Floridia
Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy
floridia@dmi.unict.it
Maria Alessandra Ragusa
Dip. di Matematica e Informatica, Università di Catania, Viale A. Doria 6, 95125 Catania, Italy
maragusa@dmi.unict.it
[Abstract-pdf]
The authors continue the study of regularity properties for solutions of elliptic systems started by M. A. Ragusa [(1) Local H\"older regularity for solutions of elliptic systems, Duke Mathematical Journal 113 (2002) 385--397; (2) Continuity of the derivatives of solutions related to elliptic equations, Proc. Royal Society of Edinburgh 136(A) (2006) 1027--1039], proving, in a bounded open set $\Omega$ of ${\mathbb R}^n$, local differentiability and partial H\"older continuity of the weak solutions $u$ of nonlinear elliptic systems of order $2m$ in divergence form \begin{equation*} \sum_{|\alpha|\leq m}(-1)^{|\alpha|} D^\alpha \, a^\alpha (x, Du) = 0. \end{equation*} Specifically, we generalize the results obtained by S. Campanato and P. Cannarsa [Differentiability and partial H\"older continuity of the solutions of nonlinear elliptic systems of order $2m$ with quadratic growth, Ann. Scuola Norm. Sup. Pisa (4)8 (1981) 285--309] under the hypothesis that the coefficients $a^\alpha (x, Du)$ are strictly monotone with nonlinearity $q = 2$.
Keywords: Higher order nonlinear elliptic systems, divergence form, monotone coefficients, generalized Sobolev spaces, local differentiability.
MSC: 35J48, 35D10; 35J45, 35D30
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