Plateau's problem for surfaces of prescribed mean curvature in given regions
Given some set \(K\subset {\mathbb{R}}^ 3\), a closed Jordan curve \(\Gamma\) \(\subset int K\) and some function \(H: K\to {\mathbb{R}}\) then the following Plateau type problem will be considered: Determine a surface \(X=X(u,v)\in C^ 2(B,{\mathbb{R}}^ 3)\cap C^ 0(\bar B,{\mathbb{R}}^ 3)\) which satisfies 1. \(X(\bar B)\subset K\), 2. \(\Delta X=2H(X)X_ u\bigwedge X_ v\), 3. \(| X_ u|^ 2-| X_ v|^ 2=X_ uX_ v=0\), 4. \(X_{| \partial B}: \partial B\to \Gamma\) is a homeomorphism (B denotes the open unit disc). It is well known that 2) and 3) express the fact that X has mean curvature H at points where \(| X_ u| \neq 0\). In case that \(| H|\) is smaller than the mean curvature (with respect to the inward unit normal) of \(\partial K\), various assumptions on H are available in the literature which guarantee the solvability of 1. - 4. The paper is concerned with obstacles \(\partial K\) which do not necessarily satisfy this curvature condition. To be more specific we state the main theorem: Let \(\Lambda\) denote the maximum of the absolute values of the prescribed mean curvature and the mean curvature of the boundary \(\partial K\), respectively, and let R denote the distance from \(\Gamma\) to \(\partial K\). If \[ \Lambda \leq \{-1/4R^ 2+\Pi /2c_ 0A_{\Gamma,K}\}^{1/2}-1/2R,\quad (c_ 0\cong 1.83), \] then there exists a surface X solving the Plateau problem, corresponding to 1. - 4. Here \(A_{\Gamma,K}\) denotes the infimum of area of surfaces which are spanned by \(\Gamma\) and have values in K. In addition some variants and generalizations of this theorem are proved in cases, where H satisfies certain isoperimetric conditions.
scientific article; zbMATH DE number 3944717
scientific article