Partial Hölder continuity results for solutions of non linear non variational elliptic systems with limit controlled growth

Fattorusso, Luisa ; Idone, Giovanna

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 747-754

Résumé

Let $\Omega$ be a bounded open subset of $R^{n}$, $n > 4$, of class $C^{2}$ . Let $u\in H^{2}(\Omega)$ a solution of elliptic non linear non variational system $$a(x, u, Du, H(u) )=b(x, u, Du)$$ where $a(x, u, \mu, \xi)$ and $b(x, u, \mu)$ are vectors in $R^{N}$, $N\geq 1$, measurable in $x$, continuous in $(u, \mu, \xi)$ and $(u, \mu)$ respectively. Here, we demonstrate that if $b(x, u, \mu)$ has limit controlled growth, if $a(x, u, \mu, \xi)$ is of class $C^{1}$ in $\xi$ and satisfies the Campanato condition $(A)$ and, together with $\frac{\partial a}{\partial \xi}$, certain continuity assumptions, then the vector $Du$ is partially Hölder continuous for every exponent $\alpha 1-\frac{n}{p}$.