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

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

by Michael Struwe
     
 

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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

Overview

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.

The Princeton Legacy Library uses 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.

Product Details

ISBN-13:
9780691607757
Publisher:
Princeton University Press
Publication date:
07/14/2014
Series:
Mathematical Notes Series
Pages:
158
Product dimensions:
6.00(w) x 9.10(h) x 0.40(d)

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Plateau's Problem and the Calculus of Variations


By Michael Struwe

PRINCETON UNIVERSITY PRESS

Copyright © 1989 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08510-4



CHAPTER 1

I. Existence of a solution.


1. The parametric problem. Let Γ be a Jordan curve in IRn. The "classical" problem of Plateau asks for a disc-type surface X of least area spanning Γ. Necessarily, such a surface must have mean curvature 0. If we introduce isothermal coordinates on X (assuming that such a surface exists) we may parametrize X by a function X(w) = (X1(w)1, ..., Xn(w)) over the disc

B = {w = (u, v) [member of] IR2 | u2 + v2< 1}

satisfying the following system of nonlinear differential equations

(1.1) ΔX = 0 in B1

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(1.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an (oriented) parametrization of Γ.


Here and in the following Xu = [[partial derivative] / [partial derivative]u] X and • denotes the scalar product in Euclidian IRn.

Conversely, a solution to (1.1) - (1-3) will parametrize a surface of vanishing mean curvature (away {torn branch points where [nabla] X (w) = 0) spanning the curve Γ, i.e. a surface satisfying the required boundary conditions and whose surface area is stationary in this class. Thus (1.1) - (1.3) may be considered as the Euler Lagrange equations associated with Plateau's minimization problem.

However, (1.1) - (1.3) no longer require X to be absolutely area-minimizing. Correspondingly, in general solutions to (1.1) - (1.3) may have branch points, self-intersections, and be physically unstable - properties that we would not expect to observe in the soap film experiment. Thus as we specify the topological type of the solutions and relax our notion of "minimality" a new mathematical problem with its own characteristics evolves.

In the following we simply refer to solutions of (1.1) -(1.3) as minimal surfaces spanning Γ.

In this first chapter we present the classical solution to the parametric problem (1.1) - (1.3). Later we analyze the structure of the set of Jill solutions to (1.1) -(1.3). The key to this program is a variational principle for (1.1)-(1.3) which is "equivalent" to the least area principle but is not of a physical nature as it takes account of a feature present in the mathematical model but not in the physical solution itself: The parametrization of a solution surface. This variational principle is derived in the next section. Applying the "direct methods in the calculus of variations" we then obtain a (least area) solution to the problem of Plateau. At this stage the Courant-Lebesgue-Lemma will be needed. Finally, some results on the geometric nature of (least area) solutions will be recalled.

It will often be convenient to use complex notation and to identify points w = (u, v) [member of] B with complex numbers w = u + iv = reiφ [member of] C. Moreover, we introduce the complex conjugate [bar.w] = u — iv and the complex differential operators

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that [bar.[partial derivative]][partial derivative] = Δ ; hence any solution X to (1.1) - (1-2) gives rise to a holomorphic differential [partial derivative]X : B [subset] C -> Cn satisfying the conformality relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], cp. Lemma 2.3. Conversely, from any holomorphic curve F : B [subset] C ->Cn satisfying the compatibility conditon F2 = 0 a solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to (1.1), (1.2) may be constructed.

This relation between minimal surfaces and holomorphic curves is the basis for the classical Weierstraß - Enneper representations of minimal surfaces in IR3 which constitute one of the major tools for constructing and investigating minimal surfaces, cp. Nitsche [1, §§ 155 - 160].

2. A variational principle. Let H1,2 (B;IRn) be the Sobolev space of L2 — functions X : B ->IRn with square integrable distributional derivatives, and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

denote L2 the respectively the seminorm and norm in H1,2 (B; IRn)

For X [member of] H1,2 (B; IRn) let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

denote the area of the "surface" X, cp. Simon [1, p. 46].

Also introduce the class

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a weakly monotone parametrization of ofΓ}

of H1,2 surfaces spanning Γ.

Note that the area of a surface X does not depend upon the parametric representation of X, i.e.

(2.1) A(X ο g) = A (X)

for all diffeomorphisms g of [bar.B]. Hence by means of the area functional it is impossible to distinguish a particular parametrization of a surface X, and any attempt to approach the Plateau problem by minimizing A over the class C(Γ)is doomed to fail due to lack of compactness.

In 1930/31 3esse Douglas and Tibor Radó however ingenuously proposed a different variational principle where the minimization-method meets success: They (essentially) considered Dirichlet's integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

instead of A. For this functioned the group of symmetries is considerably smaller; the relation

(2.2) D(X ο g) = D(X)

only holds for conformal diffeomorphisms g of [bar.B], i.e. for diffeomorphisms g satisfying the condition

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, A and D are related its follows: For X [member of] H1,2 (B;IRn)

(2.4) A (X) ≤ D (X)

with equality if X is conformal, i.e. satisfies (1.2).

Conversely, given a surface parametrized by X [member of] H1,2 (B;IRn) we can assert the following result due to Morrey [2; Theorem 1.2]:

Theorem 2.1: Let X [member of] H1,2 (B;IRn), ε > 0. There exists a diffeomorphism g : B ->B such that X' = X ο g satisfies:

D (X') ≤ (1 + ε) A (X') = (1 + ε) A (X).


In particular, Theorem 2.1 implies that

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We will not prove Morrey's ε-conformality result. However, with the tools developed in Chapter 4 it will be easy to establish (2.5) for rectifiable Γ, cp. the appendix.

By (2.5), for the purpose of minimizing the area among surfaces in C(Γ) it is sufficient to minimize Dirichlet's integral in this class. Moreover, we have the following

Lemma 2.2:X [member of] C(Γ) solves the Plateau problem (1.1) - (1.3) iff X is critical for D on C(Γ) in the sense that

i) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any family of diffeomorphisms gε : [bar.B] -> [bar.B]ε depending differentiably on a parameter [absolute value of ε] < ε0, and with gο = id.

Proof: Compute

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence the first stationarity condition i) is equivalent to the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which in turn is just the weak form of the differential equation (1.1). By standard regularity results any weak solution of X [member of] H1,2 (B;IRn) of (1.1) will be smooth in B and (1.1) will be satisfied in the classical sense.

It remains to show that for harmonic X [member of] C(Γ) the stationarity condition ii) is equivalent to the conformality relations (1.2). This result requires some preparatory lemmata which we state in a slightly more general way than will actually be needed.

Lemma 2.3: Let G be a domain in IR2 = C and suppose X [member of] H1,2 (G;IRn) is harmonic. Then the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a holomorphic function of w = u + iv [member of] G [subset] C.

Proof: Note that Φ may be written as a product

Φ = (Xu - iXv)2 = ([partial derivative]X)2

with component-wise complex multiplication and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the usual complex differential operators.

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence by harmonicity of X

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e. Φ is holomorphic.


Lemma 2.4: Suppose G is a domain in IR2 and let X [member of] H1,2 (G;IRn). Moreover, suppose that for any differentiable family of diffeomorhisms gε : [bar.G] -> [bar.G]εg]omni] = id there holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then X is conformal.

Proof: Let τ [member of] C1 ([bar.G];IR2) and for with ε [member of] IR with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

consider maps gε = id + ετ : G ->Gε := gε (G) Since by choice of the maps gε are injective and the rank of the differential

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is maximal everywhere the gε in fact are diffeomorphisms gε: [bar.G] -> [bar.G]ε.

Compute by the chain rule:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]


while - labeling [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

I.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is now clear that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is differentiable at and ε = 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].


If now again we consider IR2 [??] C by letting w = u + iv, τ = τ1 + iτ2 we may rewrite the integrand as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where Φ is defined as in Lemma 2.3.

Thus

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the expression can only vanish for all τ [member of] C1([bar.G]; IR2) if Φ vanishes identically in G, i.e. if X is conformal.

To conclude the proof of Lemma 2.2 in view of Lemma 2.4 it suffices to remark that by (2.6) conformality of X also implies the stationarity condition ii) of Lemma 2.2. Hence the critical points of D in C(Γ) precisely correspond to the solutions of Plateau's problem.

Remarks 2.5. i) If X is harmonic on B, by Lemma 2.3 and upon integrating by parts in (2.6) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the conformality relations (1.2) may be interpreted as a natural boundary condition for the holomorphic function Φ associated with X. Cp. Courant [1, p. 72 ff].

ii) Variations of the type i) in Lemma 2.1 may be interpreted as "variations of the dependent variables" i.e. of the surface X. Variations of the type ii) ("variations of the independent variables") correspond to variations of the parametrization of X.

iii) By conformal invariance of D and the Riemann mapping theorem any minimizer X0 of D in C (r) will be a critical point of D in the sense of Lemma 2.1. Indeed, by (2.6) it suffices to show that X0 satisfies the stationarity condition ii) of Lemma 2.2 for all gε = id + ετ, τ [member of] C1 ([bar.B]; IR2).

Suppose by contradiction that for some τ [member of] C1 ([bar.B]; IR2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with Bε (id + ε tau])(B), Then for some and ε ≠ 0 and Xε = X0 ο (id + ετ)-1 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But Bε is conformed to B. Hence we may compose Xε with a conformal map gε : [bar.B] -> [bar.B]ε to obtain a comparison surface ??ε = Xε ο gε [member of] C(Γ) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The contradiction proves that X0 is critical for D.

3. The direct methods in the calculus of variations. We now proceed to derive the existence of a minimizer of D on C(Γ)- and hence of a solution to Plateau's problem (1.1) - (1.3), cp. Remark 2.4. iii) - from the following general principle:

Theorem 3.1: Let M be a topological Hausdorff space, and let E : M ->IR [union] {∞}.

Suppose that for any the set α [member of] IR the set

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is compact.


(Continues...)

Excerpted from Plateau's Problem and the Calculus of Variations by Michael Struwe. Copyright © 1989 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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