On the Plateau problem for surfaces with minimum moment of inertia

1 Introduction

Let α ∈ ℝ be a real number. Consider a smooth orientable surface M immersed in Euclidean space ℝ 3 . Define the energy functional

where dM stands for the induced area element. This energy extends, in dimension two, the moment of inertia of planar curves γ ⁢ ( t ) = ( x ⁢ ( t ) , y ⁢ ( t ) ) , t ∈ [ a , b ] ,

for curves γ satisfying γ ⁢ ( a ) = P 1 and γ ⁢ ( b ) = P 2 , where P 1 , P 2 are two given points of the plane ℝ 2 . Critical points of this functional were studied by Euler in a classical treatise [9]. Introducing polar coordinates r and θ, all regular extremals of ℐ α are described by the relation

while also “discontinuous solutions” composed of the straight lines θ = θ 1 = arg ⁢ P 1 and θ = θ 2 = arg ⁢ P 2 connecting P 1 and P 2 and intersecting at the origin, can be observed, see Carathéodory [2, pp. 307–308], Tonelli [23, p. 340] and in particular Mason [20]. Mason proved that every regular extremal (1.1) joining two points P 1 = ( r 1 , θ 1 ) and P 2 = ( r 2 , θ 2 ) , where | θ 2 - θ 1 | < π α + 1 has least integral ℐ α among all curves joining P 1 and P 2 , while if | θ 2 - θ 1 | ≥ π α + 1 only the discontinuous extremal formed by the straight lines θ = θ 1 , θ = θ 2 minimizes ℐ α . In fact, there is no regular extremal (1.1) which connects P 1 and P 2 in this case.

The n-dimensional analogue of ℐ α has only recently found some interest (see [7, 4, 3]) and it was shown that critical points Σ of E α satisfy the curvature equation

where H and ν are the mean curvature (i.e., the sum of the principal curvatures) and the corresponding unit normal of ∑ at p ∈ ∑ . We henceforth call ∑ an “α-stationary surface”.

Let us introduce the variational problem

for X ⁢ ( u , v ) = ( x ⁢ ( u , v ) , y ⁢ ( u , v ) , z ⁢ ( u , v ) ) , where X u = ∂ ⁡ X ∂ ⁡ u , X v = ∂ ⁡ X ∂ ⁡ v , B = { ( u , v ) ∈ ℝ 2 : u 2 + v 2 < 1 } stands for the unit disc in ℝ 2 , and

denotes the cross product of two vectors a = ( a 1 , a 2 , a 3 ) and b = ( b 1 , b 2 , b 3 ) in ℝ 3 . Since for conformal parameters ( u , v ) we have

while for arbitrary parameters

we replace E α by the conformally invariant Dirichlet integral

and (1.3) by the Plateau problem

where the minimum is to be taken in the Sobolev class

Here K ⊂ ℝ 3 denotes some closed set and Γ ⊂ K is a closed Jordan arc which is contained in K.

Also, let C m , β ⁢ ( B ) stand for the class of m times differentiable functions, which together with its derivatives up to m-th order satisfy a local Hölder condition with exponent β on B. Moreover, H s , loc m ⁢ ( B ) stands for the Sobolev class of m times weakly differentiable functions, which are all locally integrable on B to the exponent s ≥ 1 .

Note that for bounded K ⊂ ℝ 3 we obviously have D α ⁢ ( X ) < ∞ for all X ∈ C ⁢ ( Γ , K ) and α ≥ 0 . The same conclusion holds for negative α if 0 ∉ K .

In this paper we prove the following existence result for the Plateau problem P α .

Theorem 1 will be proved in Section. 3.

In Section 4 we will also derive some enclosure results for α-stationary surfaces. These results can be viewed as a generalization of the well-known convex hull property of minimal surfaces. In addition, we prove non-existence of multiply-connected α-stationary surfaces if the boundary has values in the two half cones of a given cone of ℝ 3 . This extends a well-known result for minimal surfaces [15] to α-stationary surfaces.

Similarly, we obtain more enclosure results for α-stationary surfaces having several boundary components if each one of these components is contained in a suitable set K ⊂ ℝ 3 . In this paper we obtain enclosure results when K is a hyperboloid (Theorem 4) or a halfspace, a cylinder, a ball or a complement of a ball (Theorem 5). In all of them, the value of α will depend on the particular domain K. The proofs are based on the maximum principle for subsolutions of certain elliptic equation of second order.

In Section 2 we give some preliminaries, in particular explicit examples of α-stationary surfaces and we will see that solutions of the system (1.5)–(1.6) are α-stationary surfaces, that is, satisfy the curvature equation (1.2). As a consequence of Theorem 1 we hence obtain the existence result Theorem 3 for the Plateau problem for α-stationary surfaces if the boundary curve Γ fulfills the requirements of Theorem 1. As can already be seen from the one-dimensional case, these assumptions are natural, and – up to quantitative improvements – also necessary.

2 Preliminaries and existence of α-stationary surfaces

The energy E α contains the distance | p | of a point to the origin 0 ∈ ℝ 3 . This provides a relevant role to that point. For example, E α is in general not differentiable if 0 ∈ M . This also affects the behavior of stationary surfaces with respect to transformations of ℝ 3 . For example, if M is an α-stationary surface and A : ℝ 3 → ℝ 3 is a rigid motion of ℝ 3 , then the surface A ⁢ ( M ) is not, in general, an α-stationary surface because rigid motions do not preserve the denominator of (1.2). This is the case for translations of ℝ 3 . However, it is immediate that if A is a vector isometry, that is, if A ∈ O ⁢ ( 3 ) is an orthogonal matrix, then A ⁢ ( M ) is an α-stationary surface. It is also clear that dilations of ℝ 3 from 0 preserve stationary surfaces for the same value of α. As a consequence all results of this paper hold up to a vector isometry A ∈ O ⁢ ( 3 ) or up to a dilation from 0.

Examples of α-stationary surfaces can be found in the class of surfaces with constant mean curvature. It is immediate that (cp. [8, Proposition 2.3]):

1. A plane is a stationary surface if and only if it is a vector plane, i.e., contains the origin. This occurs for all α ∈ ℝ .

2. The only stationary spheres are spheres centered at 0 ( α = - 2 ) and spheres containing 0 ( α = - 4 ).

However, no cylinders are stationary surfaces. The fact that the only values of α where there exist stationary spheres are -2 and -4 gives to both numbers a prominent role in the theory of stationary surfaces. An example of this appears if we ask for those α-stationary surfaces that are closed surfaces (compact without boundary). Notice that spheres centered at 0 ( α = - 2 ) are closed but not spheres passing through 0 ( α = - 4 ). It was proved in [7, Theorem 1.6] the following result:

1. If α > - 2 , there are no closed α-stationary surfaces.

2. Let α < - 2 . If Σ is a (non-extendible) α-stationary surface, then its closure Σ ¯ must contain the origin 0 of ℝ 3 . In particular, there are no closed α-stationary surfaces.

The proofs involve computation of the Laplacian of the function | p | 2 and Hopf’s maximum principle. If α = - 2 , the same arguments prove that the only closed stationary surfaces are spheres centered at the origin.

We now make some comments concerning Plateau’s problem P α . Consider a parametrized surface X : Ω → ℝ 3 , where Ω ⊂ ℝ 2 is a domain and let w = ( u , v ) ∈ Ω , X ⁢ ( w ) = ( x ⁢ ( u , v ) , y ⁢ ( u , v ) , z ⁢ ( u , v ) ) be the expression of X in standard coordinates ( x , y , z ) of ℝ 3 .

Furthermore, let ω : ℝ 3 → ℝ denote a differentiable function. The Euler equations for the two-dimensional parametric integral

are

where

denotes the gradient of ω, and ω = ω ⁢ ( X ⁢ ( u , v ) ) , ω ξ = ω ξ ⁢ ( X ⁢ ( u , v ) ) . Assuming ω > 0 , equation (2.1) is equivalent to

If we now take

then (2.2) becomes

that is, equation (1.5).

Using the vector identity a ∧ ( b ∧ c ) = ( a ⋅ c ) ⁢ b - ( a ⋅ b ) ⁢ c for a , b , c ∈ ℝ 3 , and the conformality relations (1.6), we get

and the same computation as in [1, p. 252] shows that equation (2.2) is equivalent to the mean curvature system

where

stands for the unit normal of the parametrized surface X = X ⁢ ( u , v ) . Hence any regular conformal C 2 -solution of equation (2.2) has mean curvature

In particular, for ω ⁢ ( ξ ) = | ξ | α , regular extremals X of D α satisfy the mean curvature equation

which coincides with (1.2). Hence regular solutions of (1.5) and (1.6) are also α-stationary surfaces.

Let now Γ ⊂ ℝ 3 be a closed curve and consider mappings X such that X | ∂ B : ∂ ⁡ B → Γ is a parametrization of Γ. For conformal mappings X, we have

while in general only

holds true. In particular, this yields the inequality

and equality holds, if and only if, X is conformally parametrized. Thus we find the relation

Indeed, it follows from Morrey’s lemma on ϵ-conformal mappings, that every solution X ∈ C ⁢ ( Γ , K ) of the variational problem P α defined by

is also a minimizer for the functional E α in C ⁢ ( Γ , K ) and especially

see [21, pp. 141–143] and [22, pp. 815–817]; also in [5, p. 274]. In conclusion, we have from Theorem 1 and Remarks 1 after Theorem 1 the following theorem.

3 Proof of Theorem 1

Let α , m and R be chosen as in the theorem and put

to denote the “obstruction set” which obviously is “quasiregular” in the sense of Hildebrandt [16, 17], see also [6, p. 380]. Here ∂ ⁡ K is of class C ∞ and K is compact. Also K is contained in the half-space { ξ ∈ ℝ 3 : ξ 3 > 0 } , except for the case (ii), which is well known and will hence not be considered further. Let ω ⁢ ( ξ ) = | ξ | α ∈ C ∞ ⁢ ( K , ℝ ) . Then there are positive constants m 0 ≤ m 1 such that in all cases (i)–(iv) we have the coercivity condition

for all ( ξ , p 1 , p 2 ) ∈ K × ℝ 3 × ℝ 3 , as well as the bound

Also C ⁢ ( Γ , K ) ≠ ∅ and D α ⁢ ( X ) < ∞ for all X ∈ C ⁢ ( Γ , K ) . We consider the Plateau-obstacle problem

which belongs to a class of variational problems introduced and studied by Morrey [21, 22] and Hildebrandt [16]. For a comprehensive presentation of the theory see also the monographs [6] and [5]. We conclude from [5, p. 358] and [6, pp. 380–398] the following results:

1. P α has a solution X ∈ C ⁢ ( Γ , K ) , which satisfies the conformality relations (1.6). In addition, every solution X of P α is Hölder-continuous on B, continuous on B ¯ and maps ∂ ⁡ B monotonically onto Γ.

2. Every solution X of P α is of class H s , loc 2 ⁢ ( B , ℝ 3 ) ∩ C 1 , β ⁢ ( B ) for all s ∈ [ 1 , ∞ ) and all β ∈ [ 0 , 1 ) .

3. Let X ∈ C ⁢ ( Γ , K ) be some solution of P α and put

   J := { w ∈ B : X ⁢ ( w ) ∈ ∂ ⁡ K }

   to denote the closed “coincidence” set. Then X ∈ C 2 , β ⁢ ( B - J , ℝ 3 ) , in fact, X is analytic in B - J and satisfies the Euler equation (1.5) classically in B - J .

It remains to show that, under the assumptions of Theorem 1, the coincidence set J = { w ∈ B : X ⁢ ( w ) ∈ ∂ ⁡ K } is empty. To this end, we consider the quadratic polynomial σ : B ¯ → ℝ 3 defined by

where X ⁢ ( u , v ) = ( x ⁢ ( u , v ) , y ⁢ ( u , v ) , z ⁢ ( u , v ) ) , ( u , v ) ∈ B ¯ , is a solution of P α . Since X | ∂ B maps ∂ ⁡ B onto Γ and the curve Γ is contained in B ¯ R ⁢ ( m ¯ ) , we get σ ⁢ ( u , v ) ≤ R 2 for all ( u , v ) ∈ ∂ ⁡ B . Also we have σ ∈ H 2 , loc 2 ⁢ ( B , ℝ ) ∩ C 0 ⁢ ( B ¯ ) .

In order to prove that, in fact, σ ⁢ ( u , v ) < R 2 for all ( u , v ) ∈ B , we prove that σ is a subsolution to some elliptic operator so that the maximum principle can be applied. Define the operator

with locally bounded L 2 ⁢ ( B ) -coefficients. Note that this suffices for the applicability of the strong maximum principle [10, Theorem 9.6]. We claim that L ⁢ σ ≥ 0 almost everywhere in B and compute successively

almost everywhere in B, whence

On B - J the Euler equation (1.5) holds classically, and thus we can substitute in the above expression of L ⁢ σ the value of Δ ⁢ X given in (1.5). Also, from (1.5), taking the last coordinate of X, we have

Using also this identity, we obtain in B - J the relation

On the other hand, we obtain σ ⁢ ( u , v ) = R 2 for all ( u , v ) ∈ J , in particular σ u = σ v = Δ ⁢ σ = 0 a.e. on J. We claim that L ⁢ σ ≥ 0 a.e. on B in all cases (i)–(iv). Indeed, L ⁢ σ ≥ 0 a.e. on B if and only if

or, equivalently,

1. Case α > 0 . Inequality (3.2) holds if z ≥ α 2 + α ⁢ m (and arbitrary x , y ). Since X ⁢ ( w ) ∈ B ¯ R ⁢ ( m ¯ ) with R ≤ 2 2 + α ⁢ m , it follows that

   z ⁢ ( w ) ≥ m - 2 2 + α ⁢ m = α 2 + α ⁢ m ,

   w ∈ B ¯ and hence (3.2) holds.

2. Case α = 0 . It is immediate that (3.2) holds trivially.

3. Case - 4 ≤ α < 0 . We distinguish two subcases.

   1. Subcase - 2 ≤ α < 0 . Then 2 + α ≥ 0 and since α is negative, and z ≥ 0 , inequality (3.2) holds true for all x , y , z .

   2. Subcase - 4 ≤ α < - 2 . We have x 2 + y 2 + ( z - m ) 2 < R 2 and thus x 2 + y 2 + z 2 < R 2 - m 2 + 2 ⁢ m ⁢ z . Since α < - 2 , to prove (3.2) it is enough to check

      (3.3) - ( 2 + α ) ⁢ ( R 2 - m 2 + 2 ⁢ m ⁢ z ) ≤ - α ⁢ m ⁢ z .

      This inequality is equivalent to - ( 2 + α ) ⁢ ( R 2 - m 2 ) ≤ ( 4 + α ) ⁢ m ⁢ z , which holds trivially because R < m .

4. Case α < - 4 . It is enough to check (3.3). Using the hypothesis on R, it is sufficient that

   - ( 2 + α ) ⁢ ( R 2 - m 2 + 2 ⁢ m ⁢ z ) ≤ - ( 2 + α ) ⁢ m ⁢ ( - α ⁢ ( α + 4 ) ( 2 + α ) 2 ⁢ m + 2 ⁢ z ) ≤ - α ⁢ m ⁢ z ,

   or equivalently, after simplifications, that

   z ≤ α α + 2 ⁢ m .

   But this holds because z ≤ m + R , and using again the hypothesis on R, we get

   z ≤ m + R ≤ m - 2 2 + α ⁢ m = α α + 2 ⁢ m .

Once we have proved L ⁢ σ ≥ 0 a.e. on B, the strong maximum principle [10, Theorem 9.6], implies that X ⁢ ( B ) is completely contained in the interior of the ball B ¯ R ⁢ ( m ¯ ) . In fact, if we had σ ⁢ ( u 0 , v 0 ) = R 2 for some interior point ( u 0 , v 0 ) ∈ B , then σ ≡ R 2 on B ¯ which would be a contradiction to the assumption that Γ ∩ int ⁡ B R ⁢ ( m ¯ ) ¯ ≠ ∅ . This means that the coincidence set J is empty and hence by classical regularity theory X is an analytic solution of (1.5). This finishes the proof of Theorem 1.

4 Enclosure theorem and non-existence of multiply connected extremals

In this section, we consider solutions X ∈ C 2 ⁢ ( Ω , ℝ 3 ) ∩ C 0 ⁢ ( Ω ¯ , ℝ 3 ) of the Euler equation (1.5), which are defined on the closure of a bounded, open connected set Ω ⊂ ℝ 2 and satisfy the system (1.5)–(1.6) in Ω, where | X ⁢ ( w ) | ≠ 0 for all w ∈ Ω . We have seen above that any regular solution X satisfies the mean curvature equation

where ν = X u ∧ X v | X u ∧ X v | , and vice versa, every solution of (4.1), whose parameters are chosen conformal, fulfills equation (1.5).

The following result extends a classical enclosure theorem of surfaces with constant mean curvature [15] and it is a consequence of the conformality condition and a strong maximum principle.