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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

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)* = (*X*1(*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* = *rei*φ [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 *F*2 = 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 *IR*3 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 *L*2 — functions *X* : *B* ->*IRn* with square integrable distributional derivatives, and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

denote *L*2 the respectively the seminorm and norm in *H*1,2 (*B; IRn*)

For *X* [member of] *H*1,2 (*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 *H*1,2 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] *H*1,2 (*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] *H*1,2 (*B;IRn*) we can assert the following result due to Morrey [2; Theorem 1.2]:

**Theorem 2.1:** Let *X* [member of] *H*1,2 (*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 *g*ε : [bar.B] -> [bar.B]ε depending differentiably on a parameter [absolute value of ε] < ε0, and with *g*ο = 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] *H*1,2 (*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 *IR*2 = *C* and suppose *X* [member of] *H*1,2 (*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 *IR*2 and let *X* [member of] *H*1,2 (*G;IRn*). Moreover, suppose that for any differentiable family of diffeomorhisms *g*ε : [bar.G] -> [bar.G]ε*g]*omni] = id there holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then *X* is conformal.

**Proof:** Let τ [member of] *C*1 ([bar.G];*IR*2) and for with ε [member of] *IR* with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is maximal everywhere the *g*ε in fact are diffeomorphisms *g*ε: [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 *IR*2 [??] *C* by letting *w* = *u* + *iv*, τ = τ1 + *i*τ2 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] *C*1([bar.G]; *IR*2) 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 *X*0 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 *X*0 satisfies the stationarity condition ii) of Lemma 2.2 for all *g*ε = id + ετ, τ [member of] *C*1 ([bar.B]; *IR*2).

Suppose by contradiction that for some τ [member of] *C*1 ([bar.B]; *IR*2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The contradiction proves that *X*0 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 fromPlateau's Problem and the Calculus of VariationsbyMichael Struwe. Copyright © 1989 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

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