The Dirichlet problem for the equation of prescribed mean curvature
type: Item
instance of: scholarly article
title: The Dirichlet problem for the equation of prescribed mean curvature
zbMATH Open document ID: 0784.53001
DOI: 10.1016/S0294-1449(16)30225-6
author: Guofang Wang
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1993-06-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78293
review text: The paper deals with the Dirichlet problem of the equation (1) \(\Delta X=2H(X)X_ u \wedge X_ v\), \({X:B \to \mathbb{R}^ 3}\), \(B\) the unit disc of \(\mathbb{R}^ 2\), \(X=X(u,v)\), and \(\wedge\) the vector product of \(\mathbb{R}^ 3\). (1) is the Euler Lagrange equation of the energy functional \(E_ H(X)={1 \over 2} \int_ B \bigl( | \nabla X |^ 2+{4 \over 3}Q(X) \cdot X_ u \wedge X_ v \bigr) dudv\) with \[ Q(x_ 1,x_ 2,x_ 3)=\left( \int^{x_ 1}_ 0H(s,x_ 2,x_ 3)ds,\;\int^{x_ 2}_ 0H(x_ 1,s,x_ 3)ds,\;\int^{x_ 3}_ 0H(x_ 1,x_ 2,s)ds \right). \] Sharpening a previous result of \textit{M. Struwe} [Analysis, et cetera, Res. Pap. in Honor of J. Moser's 60th Birthd., 639-666 (1990; Zbl 0703.53049)] the author states the following theorem: Suppose that \(X_ 0 \in C^ 2(B,\mathbb{R}^ 3)\), \(X_ 0 \neq\) const, and \(H_ 0 \in \mathbb{R} \backslash \{0\}\) are such that \(E_{H_ 0}\) admits a relative minimizer in \(X_ 0+H_ 0^{1,2}\). Then there is \(\alpha>0\) such that \(E_ H\) has at least two critical points in \(X_ 0+H_ 0^{1,2}\) provided that \([H-H_ 0]<\alpha\), where \[ [H]=\sup_{x \in \mathbb{R}^ 3} \biggl( \bigl( 1+| x | \bigr) \bigl( | H(x) |+| \nabla H(x) |+| Q(x) |+| \nabla Q(x) |)\biggr) \] and \(Q\) is defined as above. It seems however to the reviewer that the proof of the above result contains a vicious circle: Lemma 3.1 is based on Lemma 3.2 which uses Theorem 4.5; the proof of Theorem 4.5 uses however formula (3.7) (the reference to (2.7) is an obvious misprint) and (3.7) requires Lemma 3.1. An analogous perturbation result with an even weaker norm than \([\cdot]\) above has recently been obtained by N. Jakobowsky (Technische Hochschule Aachen, Germany). According to recent (Feb. 1995) correspondence with the author the reviewer has been convinced that the proof of the main result is correct, in spit of the deficiency in the structure of the proof mentioned in the preceding review.
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 58E99
Mathematics Subject Classification ID: 35J50
zbMATH DE Number: 220983
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: Euler Lagrange equation
zbMATH Keywords: energy functional
reviewed by: F. Tomi
MaRDI profile type: Publication
cites work: Multiple solutions ofH-systems and Rellich's conjecture
cites work: Convergence of solutions of H-systems or how to blow bubbles
cites work: Q3762950
cites work: On the plateau problem for surfaces of constant mean curvature
cites work: Randwertprobleme für Flächen mit vorgeschriebener mittlerer Krümmung und Anwendungen auf die Kapillaritätstheorie. I: Fest vorgegebener Rand
cites work: Convergence of surfaces of prescribed mean curvature
cites work: On the existence of surfaces with prescribed mean curvature and boundary
cites work: On the nonuniqueness of surfaces with constant mean curvature spanning a given contour
cites work: Q4692400
cites work: Nonuniqueness in the Plateau problem for surfaces of constant mean curvature
cites work: Large H-surfaces via the mountain-pass-lemma
cites work: Q3482498
cites work: An existence theorem for surfaces of constant mean curvature
cites work: A general existence theorem for surfaces of constant mean curvature
description: scientific article; zbMATH DE number 220983