Differentiability almost everywhere of solutions of elliptic equations
Let \(\Omega\) be an open subset of \(R^ n\). The author considers the equation \[ (1)\quad div A(x,u,\nabla u)=B(x,u,\nabla u),\quad x\in \Omega, \] with A and B subjected to the conditions: A(x,u,p) and B(x,u,p) are measurable functions whatever be measurable u and p and \[ | A| \leq a| p|^{\alpha -1}+b| u|^{\alpha - 1}+e,\quad | B| \leq c| p|^{\alpha -1}+d| u|^{\alpha -1}+f,\quad pA\geq | p|^{\alpha}-d| u|^{\alpha}-g, \] where \(a>0\), \(1<\alpha \leq n\), b, c, d, e, f, g are measurable, \(b,e\in L_{n/(\alpha -1-\epsilon)}\), \(c\in L_{n/(1- \epsilon)}\), \(d,f,g\in L_{n/(\alpha -\epsilon)}\), \(0<\epsilon <\min \{1,\alpha -1\}\). It is proved that the generalized solution of (1) in the class \(W^ 1_{\alpha,loc}(\Omega)\) is differentiable almost everywhere in \(\Omega\).
scientific article; zbMATH DE number 4113115