Removable singularities of solutions to elliptic Monge-Ampère equations
Let \(\Omega\) be a disc with center \((x_ 0,y_ 0)\). We consider a solution \(z\in C^ 2(\Omega \setminus \{(x_ 0,y_ 0)\})\) of the Monge-Ampère equation \(Az_{xx}+2Bz_{xy}+Cz_{yy}+z_{xx}z_{yy}- z^ 2_{xy}=E\). Suppose the coefficients A, B, C, and E are sufficiently regular functions of the independent variables x and y in \(\Omega\), satisfying the ellipticity condition \(AC-B^ 2+E\geq const>0\). Then it is possible to extend z to a twice continuously differentiable function in \(\Omega\) provided that one of the first derivatives \(z_ x\) or \(z_ y\) has a continuous extension to the center \((x_ 0,y_ 0)\). The proof is based on the existence of a diffeomorphism \(\Phi\) : \(\Omega\setminus \{(x_ 0,y_ 0)\}\to {\mathcal D}\), (x,y)\(\mapsto (u(x,y),v(x,y))\), where \({\mathcal D}\) is either the punctured unit circle or the punctured plane or a non-degenerate annulus and u and v solve a certain Beltrami system. This leads to a characteristic system of partial differential equations associated with the Monge-Ampère equation. The main idea now in the first two alternatives for the domain \({\mathcal D}\) is a transformation of the solution z which allows us to treat this system as if it were quasiconformal. As a consequence we obtain continuous differentiability for the functions \(z_ x\circ \Phi^{-1}\), \(z_ y\circ \Phi^{-1}\). The assertion \(z\in C^ 2(\Omega)\) now follows from the non-vanishing of the Jacobian \(\partial (x,y)/\partial (u,v)\) at the point \(\Phi (x_ 0,y_ 0)\). The third alternative is ruled out by assumption of continuity of \(z_ x\) or \(z_ y\) at the center \((x_ 0,y_ 0)\).
scientific article; zbMATH DE number 4200701
scientific article