Solutions to \(H\)-systems by topological and iterative methods
The authors consider the Dirichlet problem in a bounded \(C^{1,1}\) domain \(\Omega\subset\mathbb R^2\) for a vector function \(X:\overline{\Omega}\to\mathbb R^3\) which satisfies the equation of prescribed mean curvature \[ \Delta X= 2H(u,\nu,X)X_u\wedge X_\nu \quad\text{in }\Omega, \qquad X=g \quad\text{on }\partial\Omega, \tag{1} \] where \(\wedge\) denotes the exterior product in \(\mathbb R^3\), \(H:\overline{\Omega}\times\mathbb R^3\to\mathbb R\) is a given continuous function, and the boundary data \(g\) is smooth. Under some natural assumptions on \(g\) and \(H\), the authors show, a map generated by (1) is an open map, that is if (1) admits a solution for some \((H_0,g_0)\), then (1) is solvable for any \((H,g)\) close enough to \((H_0,g_0)\) as well. To this end, they use a Newton iteration.
scientific article; zbMATH DE number 1929997