Isolated singularities of elliptic Monge-Ampère equations in dimensions \(n\geq 2\)
Let \(u\) be a twice continuously differentiable solution of the Monge- Ampère equation \[ \text{det } D^ 2 u(x)= f(x,u(x), Du(x)) \quad \text{ in the domain } \Omega=\{x\in \mathbb{R}^ n:\;0<| x-x_ 0|< R\},\;n\geq 2, \] where \(Du\) and \(D^ 2 u\) denote the gradient and the Hessian respectively of \(u\) and the function \(f\in C^ 2 (\Omega\cup \{x_ 0\} \times \mathbb{R}\times \mathbb{R}^ n)\) is positive. If \(D^ 2 u\) is positive definite in \(\Omega\) and \(\gamma Du\) has a continuous extension to \(\Omega\cup \{x_ 0\}\) for some unit vector \(\gamma\in \mathbb{R}^ n\), we prove that the isolated singularity of \(u\) at \(x_ 0\) is removable, i.e., \(u\) has a twice continuously differentiable extension to \(\Omega\cup \{x_ 0\}\).
scientific article; zbMATH DE number 756422