A note on Monge-Ampère equation in \(\mathbb{R}^2\)
Let \(\Omega \subset \mathbb{R}^2\) be a bounded convex domain and \(f:\mathbb{R}\rightarrow (0,\infty)\) be a smooth function. Moreover, let \(u\) be a strictly convex solution of the Monge-Ampère equation \[\text{det}(D^2 u)=f(u)\] and let \(M:\Omega \rightarrow \mathbb{R}\) be the function defined by \[M=u_{yy}u_x^2-2u_{xy}u_xu_y+u_{xx}u_y^2-2\int_{\min_\Omega u}^uf(t)dt.\] In this paper, the authors prove that: \begin{itemize} \item[1)] if \(f'\geq 0\), then \(M\) assumes its maximum value on \(\partial \Omega\); \item[2)] if \(f'\leq 0\), then \(M\) assumes its maximum value either on \(\partial \Omega\) or at a point \((x_0,y_0)\) such that \(u_x(x_0,y_0)=u_y(x_0,y_0)=0\). \end{itemize} In addiction, the authors exhibit an example which shows that the conclusion in \(1)\) may fail if the condition \(f'\geq 0\) is not satisfied. The authors also point out that when solution \(u\) satisfies the boundary condition \(u=0\) on \(\partial \Omega\), then as application of the above result, one get the inequality \[2\int_{\min_\Omega u}^uf(t)dt\leq \max_{\partial \Omega}(K|\nabla u|^3)\] where \(K\) is the curvature of \(\partial \Omega\).
scientific article; zbMATH DE number 7362295