A Priori Schranken für die Ableitungen der Lösungen gewisser elliptischer Differentialgleichungssysteme

Abstract

We study elliptic systems of the following form:

$$\partial /\partial x_\beta (a_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i} }^{\alpha \beta } (x,u)\partial u^{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i} } /\partial x_\alpha ) = f^i (x,u,\nabla u), 1 \leqslant i \leqslant N.$$

The coefficients ^αβ_i are assumed to be Dini continuous and the right hand side functions fⁱ(x,u,∇u) to grow at most quadratically with respect to |∇u|. The main result is that, assuming u to be Hölder continuous, we can estimate its first order derivatives (in the supremum norm) essentially in terms of the modulus of continuity of u and the gradient of the solution w of the system$\partial /\partial x_\beta (a_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i} }^{\alpha \beta } (x,u)\partial w^{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{i} } /\partial x_\alpha ) = 0$, with w=u on the boundary.