Plateau's Problem and the Calculus of Variations. (MN-35):

Plateau's Problem and the Calculus of Variations. (MN-35):

Michael Struwe
Copyright Date: 1989
Pages: 158
https://www.jstor.org/stable/j.ctt7zv371
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  • Book Info
    Plateau's Problem and the Calculus of Variations. (MN-35):
    Book Description:

    This book is meant to give an account of recent developments in the theory of Plateau's problem for parametric minimal surfaces and surfaces of prescribed constant mean curvature ("H-surfaces") and its analytical framework. A comprehensive overview of the classical existence and regularity theory for disc-type minimal and H-surfaces is given and recent advances toward general structure theorems concerning the existence of multiple solutions are explored in full detail.

    The book focuses on the author's derivation of the Morse-inequalities and in particular the mountain-pass-lemma of Morse-Tompkins and Shiffman for minimal surfaces and the proof of the existence of large (unstable) H-surfaces (Rellich's conjecture) due to Brezis-Coron, Steffen, and the author. Many related results are covered as well. More than the geometric aspects of Plateau's problem (which have been exhaustively covered elsewhere), the author stresses the analytic side. The emphasis lies on the variational method.

    Originally published in 1989.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-6021-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-2)
    Michael Struwe
  4. A. The “classical” Plateau problem for disctype minimal surfaces.

    • I. Existence of a solution.
      (pp. 5-32)

      Let Г be a Jordan curve inIRn. The “classical” problem of Plateau asks for a disc-type surfaceXof least area spanning Г. Necessarily, such a surface must have mean curvature 0. If we introduce isothermal coordinates onX(assuming that such a surface exists) we may parametrizeXby a function X(w) = (X¹(w),..., Xn(w)) over the disc

      $B = \left\{ {w = (u,v) \in I{R^2}|{u^2} + {v^2} < 1} \right\}$

      satisfying the following system of nonlinear differential equations

      (1.1)$\Delta X = 0$inB,

      (1.2)$|{X_u}{|^2} - |{X_v}{|^2} = 0 = {X_u} \cdot {X_v}$inB,

      (1.3)$X{|_{\partial B}}:\partial B \to \Gamma $is an (oriented) parametrization of Г.

      Here and in the following${X_u} = \frac{\partial }{{\partial u}}X$, etc., and$ \cdot $denotes the scalar product in Euclidian...

    • II. Unstable minimal surfaces
      (pp. 33-88)

      The method of gradient line deformations and the minimax-principle are the most general avaible tools for obtaining unstable critical points in the calculus of variations. Historically, the use of these methods can be traced back to the beginning of this century, cf. Birkhoff’s [1] theorem on the existence of closed geodesies on surfaces of genus 0 . Through their famous improvement of BirkhofF’s result the names of Ljusternik and Schnirelman [1] became intimately attached to these methods. In 1964 a major extension of these techniques was proposed by Palais [1], [2], Smale [1] and Palais - Smale [1]. Their fundamental...

  5. B. Surfaces of prescribed constant mean curvature.

    • III. The existence of surfaces of prescribed constant mean curvature spanning a Jordan curve in IR³.
      (pp. 91-110)

      Let Г be a Jordan curve inIR³. In part A we studied minimal surfaces spanned by Г, find we observed that any solutionXto the parametric Plateau problem (1.1.1) - (1.1.3) parametrizes a surface of vanishing mean curvature (away from branch points where$\nabla X(w) = 0$

      A natural generalization of the classical Plateau problem therefore is the following question: Given$\Gamma \subset I{R^3}$,$H \in IR$, is there a surfaceXwith mean curvatureH(for short “H-surface”) spanning Г?

      We restrict ourselves to surfaces of the type of the discB. Introducing isothermal coordinates overBon such aXwe derive the...

    • IV. Unstable H – surfaces
      (pp. 111-140)

      In the analysis of unstable minimal surfaces we relied on the existence of harmonic extensions of admissible parametrizations of Г in order to reformulate the Plateau problem in terms of a variational problem on a convex set. To imitate this procedure forH—surfaces we now considerDirichlet’s problem for the H—surface system:

      Given${X_o} \in {H^{1,2}} \cap {L^\infty }(B;I{R^3})$,$H \in IR$find$X \in {H^{1,2}} \cap {C^2}(B;I{R^3})$such that

      (1.1)$\Delta X = 2H{X_u} \wedge {X_v}$inB,

      (1.2)$X = {X_o}$on$\partial B$

      or, equivalently, find$X \in {X_o} + H_0^{1,2}(B;I{R^3})$such that

      $d{D_H}(X) = 0 \in (H_0^{1,2}(B;I{R^3})) * $

      Recall that forH= 0 the harmonic extension$\underline X $of${X_o} \in {H^{1,2}} \cap {L^\infty }(B;I{R^3})$is uniquely characterized by the relations

      (1.3)$\underline X \in {X_o} + H_0^{1,2}(B;I{R^3})$,

      (1.4)$D(\underline X ) = \inf \{ D(X)|X \in {X_o} + H_0^{1,2}(B;I{R^3})\} $

      Moreover,X...

  6. References:
    (pp. 141-148)