Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations (Q1076241)

Let u be a solution of \(\Delta u=f(x,u,\nabla u)\) in \(\Omega\), where \(\Omega\) is an open region in \(R^ m\). It is shown that the Hausdorff dimension of the singular subset \[ S=\{x\in \Omega:\quad u(x)=0\quad and\quad \nabla u(x)=0\} \] of the zero-set \(\{u=0\}\) is at most m-2. The superlinear case \(| f(x,u,\nabla u)| \leq A| u|^{\alpha}+B| \nabla u|^{\beta},\) \(\alpha\geq 1\), \(\beta\geq 1\) and the linear case are discussed separately. The main application concerns the study of the free boundary in free boundary problems.