The Douglas problem for parametric double integrals
The main result of this paper is the existence of multiply connected, conformally parametrized minimizers of genus zero for a parametric semi-elliptic double integral spanned by a given system of Jordan curves. Consider parametric functionals \({\mathcal F}\) of the form \({\mathcal F}_ B(X):=\int_ B F(X, X_ u\wedge X_ v) dudv\), where \(F \in C^ 0({\mathbb R}^ n\times{\mathbb R}^ N)\), \(n\geq 2\), \(N=n(n-1)/2\), \(B\) is a domain in \({\mathbb R}^ 2\), and \(X \in H^{1,2}(B, {\mathbb R}^ n)\cap C^ 0(\partial B, {\mathbb R}^ n)\). It is assumed that \(F\) satisfies the homogeneity condition and \(F\) is positive definite (there exist positive constants \(m_ 1\), \(m_ 2\) such that \(m_ 1| z| \leq F(x, z)\leq m_ 2| z| \) holds for all \((x, z)\in {\mathbb R}^ n\times {\mathbb R}^ N\)). Let \(\Gamma:=\langle\Gamma^ 1, \Gamma^ 2, ..., \Gamma^ k\rangle\) be a system of finitely many disjoint rectifiable closed Jordan curves in \({\mathbb R}^ n\). It is said that \(\Gamma\) satisfies the Douglas condition for \({\mathcal F}\) if and only if \(d(\Gamma)<d^{\dagger}(\Gamma)\) holds. Here, \(d(\Gamma)\) is the infimum of \({\mathcal F}_ B(X)\) among all discs \(B\) with \(k-1\) discs removed and all \(X \in H^{1,2}(B, {\mathbb R}^ n)\cap C^ 0(\partial B, {\mathbb R}^ n)\) which map \(\partial B\) onto \(\Gamma\) weakly monotonically, and \(d^{\dagger}(\Gamma)\) is the infimum of \({\mathcal F}_ B(X)\) among all disconnected compact planer regions \(B\) with \(k\) boundary components and all \(X \in H^{1,2}(B, {\mathbb R}^ n)\cap C^ 0(\partial B, {\mathbb R}^ n)\) which map \(\partial B\) onto \(\Gamma\) weakly monotonically. Under the assumption that \(F\) is semi-elliptic and \(\Gamma\) satisfies a chord-arc condition and \(F\) satisfies the Douglas condition for \({\mathcal F}\), it is proved that there exists a pair \((B, X)\) which attains \(d(\Gamma)\), and \(X\) is conformally parametrized. The main method of the proof is the Courant's classical approach for minimal surfaces described in \textit{R. Courant} [Dirichlet's principle, conformal mapping, and minimal surfaces (Interscience Publishers, New York)(1950; Zbl 0040.34603)] with the method of conformal approximation of parametric functionals which was introduced in \textit{S. Hildebrandt and H. von der Mosel} [Calc. Var. 9, 249-267 (1999; Zbl 0934.49022)]. As a by-product of the existence proof, the following isoperimetric inequality is obtained: For any pair \((B, X)\) which attains \(d(\Gamma)\), it holds that \({\mathcal A}_ B(X)\leq (m_ 2/(4\pi m_ 1))\sum_{l=1}^ k (L(\Gamma^ l))^ 2\), where \({\mathcal A}_ B(X)\) is the area functional and \(L(\Gamma^ l)\) is the length of \(\Gamma^ l\). Moreover, a regularity result is given: If \(\Gamma\) satisfies the chord-arc condition, then, for every conformally parametrized minimizer \((B, X)\) of \({\mathcal F}\), \(X\) is of class \(C^{0,\alpha}(\overline{B},{\mathbb R}^ n)\cap H_{\text{ loc}}^{1,q}(B,{\mathbb R}^ n)\) for some \(\alpha\in(0,1/2]\) and \(q>2\). This result follows from corresponding result for the Plateau problem proved in \textit{S. Hildebrandt and H. von der Mosel} [op. cit.].
scientific article; zbMATH DE number 1889223