Local solutions to a class of Monge-Ampère equations of mixed type
The paper presents a study of Monge-Ampère type equations which is motivated mainly by the geometric question of the existence of surfaces with prescribed-Gauss curvature. The general Monge-Ampère type equation is \[ {1\over \det(g)} \text{det}(\nabla^2_g u)= K(x, u,\nabla_g u),\tag{1} \] where \(g\) is Riemannian metric in some domain \(\Omega\subset\mathbb{R}^n\) and \(K\) is sufficiently smooth function in \(\Omega\times\mathbb{R}\times \mathbb{R}^n\). Equation (1) can also be viewed as special case of \(\sigma_k\)-equations. Instead of the determinant, it is possible to consider the \(\sigma_k\), the \(k\)th symmetric function of eigenvalues of the Hessian in the left-hand side of (1), as the determinant is the \(\sigma_n\). To present the idea clearly, equation (1) is considered corresponding only to the case when \(g\) is the trivial metric, that is, \(\text{det}(D^2 u)= K(x,u,Du)\) and for a class \(K\) of mixed sign, i.e. the equation \[ \text{det}(D^2 u)= \eta(\varphi(x)) h(x,u,Du),\tag{2} \] where \(\eta= \eta(t)\) is a suffciently smooth, strictly increasing function with \(\eta(0)= 0\), \(\varphi\) is a sufficiently smooth function in a neighborhood \(\Omega\subset\mathbb{R}^n\) of the origin satisfying \(\varphi(0)= 0\), \(D_\varphi(0)\neq 0\) and \(h:\Omega\times\mathbb{R}\times \mathbb{R}^n\to\mathbb{R}\) is a sufficiently smooth function with \(h(0,0,0)\neq 0\). The existence of a local solution in a neighborhood of \(0\in\mathbb{R}^n\) is under consideration. A typical example for \(\eta\) is given by \(\eta(t)= t^q\) for some odd integer \(q\) . The implicit function theorem implies that \(\varphi^{-1}(0)\) is an hypersurface in a neighbood of the origin. An example is given by \(\varphi(x)= x_n\). In general, there exists a hypersurface in a neighborhood of the origin given by \(\{\varphi= 0\}\) such that (2) is of mixed type, elliptic in one side of this hypersurface, where \(\varphi> 0\), and hyperbolic in another side, where \(\varphi< 0\). The following theorem is the main result of this article: Let \(\eta\), \(\varphi\), and \(h\) be \(C^r\)-functions, in the argument's satisfying the written above conditions, and let \(h(0,0,0,0)\neq 0\) with \(r\geq 2[n/2]+ 20\) for some integer \(r\). Then the equation (2) admits a \(C^{r-[n/2]- 15}\)-solution in some neighborhood of \(0\in\mathbb{R}^n\).
scientific article; zbMATH DE number 5132698