Existence and uniqueness of \(H\)-system's solutions with Dirichlet conditions
Let \(B= \{(u, v)\in \mathbb{R}^2\mid u^2+ v^2< 1\}\) be the unit disc. The main purpose of this paper is to find a vector function \(u:\overline B\to \mathbb{R}^3\) satisfying the prescribed mean curvature equation \(\Delta X= 2H(u,v) X_u\wedge X_v\) in \(B\), and \(X= g\) on \(\partial B\), where \(\wedge\) denotes the exterior product and \(H:\overline B\to\mathbb{R}\) is a given continuous function, while \(g\) is in a Sobolev space \(W^{2,p}(B, \mathbb{R}^3)\). Using Sobolev immersion the authors transform the above problem into a contraction mapping on the Sobolev space \(W^{1,p}\), and then derive the existence and local uniqueness for a solution of the mean curvature equation.
scientific article; zbMATH DE number 1513367