Solutions of \(H\)-systems using the Green function
The authors consider the Dirichlet problem for the prescribed mean curvature system \[ \Delta X = 2H(u,v,X)X_u\wedge X_v \quad\text{in}\quad \Omega, \qquad X = g \quad\text{on}\quad \partial\Omega,\tag{*} \] for a vector function \(X:\Omega\rightarrow {\mathbb R}^3\), where \(\Omega\) is a smooth bounded domain in \({\mathbb R}^2\). This problem and the general Plateau problem with \(H=\)constant and \(H=H(X)\) have been studied by several authors using variational methods. The present authors have used topological methods to study \((*)\) in the case \(H=H(u,v)\) [Nonlinear Anal., Theory Methods, Appl. 42 A, No. 4, 673-677 (2000; Zbl 0968.63006)]. Here a solution is found using the Green representation formula. In addition, the authors show that if there is a solution for \(H_0,g_0\), then there is also a solution for \(H,g\) sufficiently close to \(H_0,g_0\) in a suitable sense.
scientific article; zbMATH DE number 1563220