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global differentiability of weak solutions; elliptic problems; controlled growth; nonlinearity with $q=2$
Let $\Omega $ be a bounded open subset of $\mathbb{R}^{n}$, $n>2$. In $\Omega $ we deduce the global differentiability result \[ u \in H^{2}(\Omega , \mathbb{R}^{N}) \] for the solutions $u \in H^{1}(\Omega , \mathbb{R}^{n})$ of the Dirichlet problem \[ u-g \in H^{1}_{0}(\Omega , \mathbb{R}^{N}), -\sum _{i}D_{i}a^{i}(x,u,Du)=B_{0}(x,u,Du) \] with controlled growth and nonlinearity $q=2$. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.
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