The Dirichlet problem for the equation of prescribed mean curvature
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.
scientific article; zbMATH DE number 220983
scientific article