Boundedness and differentiability for nonlinear elliptic systems

Abstract:

We consider the elliptic system $\operatorname {div} (\mathcal {A}^j (x,u,\nabla u)) = \mathcal {B}^j (x,u,\nabla u)$, $j=1,\ldots ,N,$ and an obstacle problem for a similar system of variational inequalities. The functions $\mathcal {A}^j$ and $\mathcal {B}^j$ satisfy certain ellipticity and boundedness conditions with a $p$-admissible weight $w$ and exponent $1<p\le 2$. The growth of $\mathcal {B}^j$ in $|\nabla u|$ and $|u|$ is of order $p-1$. We show that weak solutions of the above systems are locally bounded and differentiable almost everywhere in the classical sense.