The behaviour of solutions to elliptic Monge-Ampère equations at singular points
Let \(\Omega \subset \mathbb{R}^ 2\) be a disc with center \((x_ 0, y_ 0)\) and \(z \in C^ 2 (\Omega \backslash\{(x_ 0, y_ 0)\})\) a solution of the elliptic Monge-Ampère equation \(Az_{xx} + 2Bz_{xy} + Cz_{yy} + z_{xx} z_{yy} - z^ 2_{xy} = E\). Assume that \(A,B,C\), and \(E\) are of class \(C^{1, \mu} (\Omega)\) for some \(\mu \in (0,1)\) and satisfy \(AC - B^ 2 + E > 0\) in \(\Omega\). This paper is concerned with the behaviour of the solution \(z\) at the isolated singular point \((x_ 0, y_ 0)\). The result is that either the singularity is removable, i.e., \(z\) has a twice continuously differentiable extension to \((x_ 0, y_ 0)\), or \(\lim_{(x,y) \to (x_ 0, y_ 0)} | z_{xx} (x,y) + z_{yy} (x,y) | = \infty\). In the latter case the radial limits of \(z_ x (x_ 0 + \tau \cos \alpha\), \(y_ 0 + \tau \sin \alpha)\) and \(z_ y (x_ 0 + \tau \cos \alpha\), \(y_ 0 + \tau \sin \alpha)\), as \(\tau \downarrow 0\), are continuous, non-constant functions of \(\alpha\).
scientific article; zbMATH DE number 704916