Computable Constants for Korn’s Inequalities on Riemannian Manifolds

A method is presented for the explicit construction of the non-dimensional constant occurring in Korn’s inequalities for a bounded two-dimensional Riemannian differentiable simply connected manifold subject to Dirichlet boundary conditions. The method is illustrated by application to the spherical cap and minimal surface.


Introduction
Korn's inequalities, first formulated in 1906 [26,27] and further treated by Friedrichs in 1947 [15], have attracted an extensive literature comprehensively reviewed in surveys that include those by Horgan [20], Gurtin [18], Kondratiev and Oleinik [23,24], and Ciarlet [9,Chap. 6]. Nonlinear versions, partly in the form of rigidity theorems, have been developed by Frieseke, James, and Müller [16], and by Ciarlet, Gratie and Mardare [10], amongst others. The main application of both linear and nonlinear inequalities is to proofs of existence, uniqueness, and continuous data dependence of solutions to boundary value problems in continuum mechanics and related theories.
The inequalities hold for bounded and unbounded regions of Euclidean space IR n , n = 2, 3, and interrelate the gradient ∇ u of a suitably differentiable vector field u and its symmetric and antisymmetric parts S, A given by R.J. Knops r.j.knops@hw.ac.uk 1 The Maxwell Institute of Mathematical Sciences and School of Mathematical and Computing Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, UK and a superscript T denotes transposition. For bounded regions ⊂ IR n , the first Korn's inequality is of the form |∇ u| 2 dx ≤ C |S| 2 dx, (1.2) in which C is a non-dimensional positive constant dependent upon the geometry of . Its optimum value is usually known as Korn's constant, but this term is conveniently used here to signify all constants C that generically feature in Korn's inequalities. Insertion of the decomposition (1.1) into (1.2) leads to the sequence which shows that C ≥ 1. Inequality (1.3) also implies both the second Korn's inequality (see Friedrichs [15]) (1.4) and the third Korn's inequality The vector field u in these inequalities must satisfy conditions apart from differentiability. For example, the first Korn's inequality is invalidated when u is a rigid body rotation as then S, but not ∇ u, identically vanishes. Various normalisations must be introduced, or the inequalities modified, to account for rigid body displacements and in particular pure rotations. This is achieved here and throughout by confining attention to bounded regions and to admissible functions that vanish on the boundary. Subject to these restrictions, the divergence theorem yields |tr ∇ u| 2 dx + |∇u| 2 dx = 2 |S| 2 dx, (1.6) where tr denotes the trace operator. We immediately conclude that and therefore (1.2) is established with C = 2, which, as remarked by Horgan [20], is the best possible constant. Other inequalities are obtained by the alternate elimination of |S| 2 and |∇ u| 2 between (1.3) and (1.6) but do not improve (1.4) and (1.5) when C = 2. For example, elimination of |S| 2 (cp, [18, p. 38]) gives  (1.10) where denotes to the first eigenvalue for the fixed membrane problem on . In conjunction with inequality (1.2), we then have (1.11) Section 2 introduces notation and basic notions from differential geometry. Section 3 is devoted to three versions of Poincaré's inequality, while Sect. 4 describes how the Korn's inequalities are constructed for the general manifold and the corresponding Korn's constants explicitly estimated. The spherical cap and minimal surface are considered in Sect. 5. Section 6 contains brief concluding observations.

Notation and Basic Differentiable Geometry
Elements of differential geometry used later are summarised in this section and serve to introduce notation. Standard textbooks, for example [4,29,32,44] and more recently [7][8][9], may be consulted for complete treatments. The convention is adopted of summation over repeated indices with Greek indices ranging over 1, 2 while Latin indices have the range 1, 2, 3. The comma notation indicates covariant differentiation and not partial differentiation. Scalar, vector and tensor quantities are not typographically distinguished. Sufficient differentiability is assumed to justify all operations.
Consider a two-dimensional simply-connected bounded differentiable Riemannian manifold M embedded in an ambient three-dimensional Euclidean space IR 3 and let ∂M denote the Lipschitz smooth boundary curve of M. When specialised to be part of a spherical surface of radius r, M is denoted by (r) with boundary ∂ (r).
Let x α , α = 1, 2, be contravariant components of the vector x that describes a curvilinear coordinate system on M, and let y ∈ M be the three dimensional position vector of a point on M. We have where x ∈ D ⊂ IR 2 , D is a bounded plane region, and e i , i = 1, 2, 3, are the orthogonal unit basis vectors of a rectangular three-dimensional Cartesian coordinate system. The map (2.1) is assumed to be an immersion. Partial differentiation with respect to the variables x α is denoted by ∂/∂x α . The covariant basis vectors in the tangent plane to M at the point y, given by a α = ∂y ∂x α , α= 1, 2, are linearly independent since the map (2.1) is an immersion. The contravariant tangent basis vectors a α , orthogonal to a α , satisfy where δ α β is the Kronecker delta of mixed order, and an interposed dot represents the scalar product.
The metric tensor, or first fundamental form, has covariant components a αβ = a α .a β = a βα , whose determinant, a, given by a = det (a αη ) = a 11 a 22 − a 2 12 > 0, is positive by linear independence of a α . Linear independence additionally implies not only the existence of a non-trivial unit normal vector a 3 = a 3 at the point y ∈ M specified by where the vector product is indicated by ×, but also that the matrix (a αβ ) with components a αβ is positive-definite in the sense that there exist non-negative constants 0 ≤ λ 0 ≤ λ 1 such that The matrix (a αβ ) is also positive-definite in the sense that A vector u in the tangent plane to M at the point y(x) has the representation u = u α a α = u α a α , (2.7) in terms of the contravariant u α and covariant u α components. It follows that indices may be lowered or raised on multiplication by a αβ and a αβ . Covariant differentiation of the covariant and contravariant components of u, indicated by subscript comma, is defined respectively to be where σ αβ , the non-tensorial Christoffel symbols of the second kind, are obtained from the Gauss relations in which b α β = a αλ b λβ , and b αβ are the symmetric components of a covariant tensor whose related quadratic form is the second fundamental form. Unlike the first fundamental form, it is not necessarily positive-definite.
Most partial differentiation operations apply to covariant differentiation and are facilitated by Ricci's lemma which states that a αβ,γ = a αβ ,γ = a ,γ = 0. (2.11) The second covariant derivatives, however, do not commute but satisfy the Ricci identity 14) 15) in terms of which the Gaussian curvature, K, of M is expressed as (2.16) The mean curvature, H , specifically of the two-dimensional manifold M ⊂ IR 3 , is defined by where, as previously stated, b α β = a αγ b γβ .

Remark 2.1
When the manifold is a plane region in IR 2 , we may select x α = y α and x 3 = y 3 = 0 to show that partial and covariant differentiation coincide and that Certain integrals that frequently recur in the following sections are listed here for reference. They involve the covariant and contravariant components of the symmetric linear strain tensor and antisymmetric rotation tensor defined by
For plane regions ⊂ IR 2 , the integrals reduce to We conclude this section by recording the form of the divergence theorem appropriate to a bounded manifold M. Let w α be the contravariant components of a differentiable surface vector w defined on M. Then M w α ,α dS = ∂M w α n α ds, (2.27) where n α are the contravariant components of the unit outward normal to the boundary curve ∂M, ds is the element of its length, and dS is given by (2.21).

Poincaré Inequalities
We sketch proofs based on variational arguments of three inequalities that are required subsequently but which are of general importance in the geometry and analysis of manifolds.

First Poincaré Inequality
The following lemma corresponds to inequality (1.10).

Lemma 3.1 (Poincaré's inequality for scalar functions) Let φ(x) be a differentiable scalar invariant function defined at each point
where the tangential surface gradient ∇ s φ on M is given by and , a positive eigenvalue of the corresponding self-adjoint elliptic linear Laplace-Beltrami differential operator L specified by

5)
and is dependent upon the geometry of M and the boundary condition (3.1).

Proof Write inequality (3.2) as the Rayleigh quotient
Standard variational arguments then show that the minimum for (3.6) subject to boundary conditions (3.1) satisfies the Euler-Lagrange equations (3.5). Consequently, and φ(x) are the eigenvalues and eigenfunctions for the Laplace-Beltrami differential operator L on the plane bounded region D ⊂ IR 2 .
Exact expressions for the eigenvalue are known for only certain geometries, and under various conditions. See, for example Grigor'yan [17], Alencar and Neto [1], Li [31], and Saloff-Coste [42]. By contrast, lower bounds for are comparatively abundant and become significant in Sect. 4 and Sect. 5 for explicit estimates of Korn's constants. A useful result is that decreases as |M| increases for homogeneous Dirichlet boundary conditions. Moreover, Sperb [45], on improving a lower bound by Cheeger [5], shows that where |∂M| and |M| are the length and surface area of the bounding curve ∂M and surface M respectively. Other lower bounds employed subsequently are described by Protter and Weinberger [39], based upon the maximum principle, and by Bandle [3] that include those exploiting the equivalence of (3.5) to the inhomogeneous fixed membrane problem. Of course, when M is a plane region , a lower bound is provided by the Faber-Krahn estimate: where j 0 ≈ 2.4048 is the smallest positive zero of the Bessel function J 0 . Equality holds for the circular disc. For our general discussion, however, we assume that either or a lower bound are explicitly known.
The next result extends the first Poincaré's inequality to surface tangent vectors.

Proposition 3.1 (A Poincaré inequality for surface vectors) Let u(x) be a smooth tangential surface vector at y(x) ∈ M that satisfies the boundary condition
where is the constant appearing in inequality (3.2).
Recall that covariant differentiation is denoted by subscript comma.

Remark 3.1
In terms of the integrals (2.22)-(2.25), inequality (3.10) is expressed as Let v i (x) be the components of the surface vector u(x) with respect to an orthogonal Cartesian set of axes with unit basis vectors e i , i = 1, 2, 3. We have the identity The Poincaré inequality (3.2) applied separately to each component v i leads to which is the desired inequality (3.10). The derivation has used expressions (2.10) for the partial derivative of a basis vector a α and (2.8) for covariant differentiation.

Remark 3.2
Under certain conditions the last term on the right of (3.10) may be removed. From (2.3) and Schwarz's inequality we successively have and therefore (3.14) Consequently, for manifolds that satisfy both N < ∞ and the condition which may be equivalently written as (3.17)

Second Poincaré Inequality
For certain manifolds, for example the hemispherical surface, inequality (3.10) becomes inappropriate when computing the corresponding Korn's constant and must be replaced. The alternative Poincaré inequality, which also holds generally, involves only the first term on the right of (3.10). We have or

19)
where the constant is the eigenvalue occurring in inequality (3.2).
Proof As before, regard inequality (3.18) as a Rayleigh quotient and apply variational arguments to conclude that the Euler-Lagrange equations consist of the uncoupled elliptic system u α + a βσ u α,σβ = 0, (3.20) or Properties of the corresponding eigenvalues and eigenfunctions consequently are those for the Laplace-Beltrami operator (3.4).

Basic Korn's Inequality on M
The divergence theorem (2.27) and boundary condition (4.2) then lead to which may be written Insertion of (4.5) into the integrated version of (4.3), re-arranged as recovers the required Korn's inequality (4.1) analogous to (1.2).

Remark 4.1 (Non-negative I 2 )
Note that I 2 ≥ 0 on manifolds M for which the Gaussian curvature K is everywhere non-positive.

Other Korn's Inequalities
Abbreviated forms of the basic Korn's inequality (4.1), the subsidiary inequality (4.5), and the identity (4.7) are from the last of which we have Other forms of Korn's inequalities for general M are now easily derived. Elimination of I 1 between (4.10) and either (3.11) or (3.19) respectively yields and Subject to boundedness assumptions (3.13) and (3.15), we conclude from (3.14) and (3.17) that On the other hand, we always have  allow Korn's inequalities (4.12) and (4.13) to be written which corresponds to (1.11). Moreover, (4.8) implies (4.21) analogous to the first Korn's inequality (1.2).

Second and Third Korn's Inequalities. Friedrich's Inequality
We complete the extension of Korn's inequalities by considering types corresponding to (1.4) and (1.5).
The rotation of the surface tangential vector u(x) has antisymmetric covariant components (see (2.19) 2 ) given by The last identity in combination with (4.9) and (4.10) yields  (4.29) corresponding to (1.4). Two particular geometries are considered in the next section for which explicit computations are possible.

Examples
Inequalities for general manifolds derived in Sect. 4 are now applied to two special surfaces for which the corresponding Korn's constants or their lower bounds may be explicitly computed. A detailed discussion is presented for the spherical cap and hemispherical surface as examples of certain confocal conical sections. Less extensively discussed are minimal surfaces.

Spherical Cap and Hemispherical Surface
Let (r) denote the surface of a spherical cap of radius r whose bounding curve ∂ (r) subtends an angle 2ω, 0 < ω ≤ π/2, at the centre of the corresponding sphere taken as origin of a rectangular Cartesian coordinate system y. Let (r, x α ) be spherical polar coordinates for a point y ∈ (r) so that The covariant components of the metric tensor and mixed components of the second fundamental form become The eigenvalue is determined from the associated Laplace-Beltrami equation (3.5), which in spherical surface polar coordinates becomes where φ is subject to the homogeneous boundary condition (3.1). Consider the separable solution (5.4) and put where m is some non-negative integer and a superposed prime indicates differentiation with respect to the argument of the function. Insertion into (5.3) leads to the associated Legendre equation for (x 1 ) where we have put μ = cos x 1 . When is chosen to be where P n (μ) is the Legendre polynomial of order n, and for temporary convenience, differentiation is denoted by the operator d.
The integers m, n are selected to ensure that the boundary condition (3.1) is satisfied. In particular, for given m, we choose n to be the smallest value for which The discussion is continued by separate consideration of the spherical cap and hemispherical surface.
In spherical polar coordinates, the integrals D 2 and I 3 defined in (2.22) 2 and (2.25) reduce to and consequently the augmented Korn's inequality (4.12) may be written where . (5.14) For the hemispherical surface, we have ω = π/2 and the boundary condition (5.10) becomes P n (0) = 0, which holds for all positive odd integers n. Therefore, we take n = 1. But then the computable constant C 1 in (5.14) is unbounded and (5.13) remains meaningful only for the spherical cap with 0 < ω < π/2. The difficulty is resolved by using (4.13) to obtain e αβ e αβ dS, (5.16) where . (5.17) On taking n = 1, we have C 2 = 1 for the hemispherical surface. Furthermore, since C 2 < C 1 for all positive integers n, Korn's augmented inequality (5.16) is preferable to (5.13) even for the spherical cap (0 < ω < π/2) but with n now determined from the boundary condition (5.10). Remark 5.2 While Sperb's [45] improvement (3.7) of Cheeger's lower bound for [5] may be used in (4.12), or (4.14), and yields a valid result, nevertheless when applied to the spherical cap, the Korn's constant is necessarily greater than the constants computed in either (5.13) or (5.16). The bound (4.12) remains important, however, for non-spherical surfaces.
The analogue of the first Korn's inequality is obtained by substitution of (5.11) and (5.15) in (4.8), and assumes the form where Another version is derived on noting the lower bound e αβ e αβ dS.

Remark 5.3
Families of confocal conicoids may be treated similarly provided there is a curvilinear coordinate system for which the corresponding associated Laplace-Beltrami equation admits a separable solution. and for μ = 0 the least value of n is n = 2. Korn's constant in (5.15) has the decreased value 2C 2 r 2 = 2r 2 /5.

Remark 5.5 (Families of spherical caps)
Certain problems of spatial stability on unbounded regions in IR 3 require a continuous sequence of Korn's inequalities belonging to a one parameter family of spherical caps contained in a half-space. The previous analysis may adapted to the problem. The half-space remains specified by y 3 ≥ 0, but the centres of the respective spherical caps are on the negative y 3 -axis at successively increasing distances d from the origin y = 0. Let r, the radius of the corresponding spherical cap (r), be related to d by d = λ r where λ is a fixed positive constant and 0 < λ < 1. Then (r) intersects the plane y 3 = 0 in a circle that subtends at the centre of (r) an angle ω = cos −1 λ independent of any particular (r). Korn's constant, likewise independent of (r), is determined by appropriate choice of m, ω and therefore n. For example, when ω = π/6 and m = 0, boundary condition (5.10) becomes and we take n = 4 to obtain for each r the Korn's constant 2C 2 r 2 = 2r 2 /19.

Minimal Surface
Suppose that the bounded manifold has no umbilic points and that through each point the orthogonal lines of curvature are taken to be the curvilinear coordinate directions. With respect to this coordinate system, we have b 12 = a 12 = 0, (5.26) and A different curvilinear coordinate system is adopted for the minimal surface of revolution discussed below. Further suppose that M is a minimal surface so that H = 0 and therefore Consequently, inequalities (4.12) or (4.13) lead to the augmented first Korn's inequality for a minimal surface in the form: The analogous first Korn's inequality, immediate from (4.8) and (5.29), is Nevertheless, to be specific, we examine the minimal surface that is a surface of revolution, or right helicoid. Introduce polar coordinates (x 1 , x 2 ) and represent each point y ∈ M as where p, q are prescribed positive constants. Suppose that 0 ≤ r 1 ≤ x 1 ≤ r 2 , and that the minimum height of the helicoid occurs at x 2 = 0, and its maximum height h at x 2 =x 2 .
In consequence, we have 0 ≤ x 2 ≤x 2 and h = px 2 + q. Define the positive constant m by mx 2 = 2π so that .
Furthermore, on setting we may express D 2 , I 1 , I 2 and I 3 as (5.42) where in the expression for I 1 we have used (2.2) and (5.39). We also have  It is instructive to compute a lower bound obtained from properties of separable solutions to the associated Laplace-Beltrami equation (3.5) which becomes Inspection shows that the lower bound (5.47) is superior and therefore preferable to (5.48), whereas for fixed r 1 , r 2 , p, the lower bound (5.54) is superior to both for sufficiently small (h − q). Remark 5.6 (Reduction to the plane region) When p = 0, the right helicoid reduces to the circular annulus in the plane y 3 = q and (5.52) becomes Bessel's equation. The eigenvalues for the circular annulus have been calculated, for example, by Ramm and Shivakumar [40], while for the circular disc (corresponding to r 1 = 0) the eigenvalue is the well-known Faber-Krahn expression j 2 0 /r 2 2 . However, the lower bounds (5.47), (5.48) and (5.54) become vacuous when p = 0 unless p vanishes to order (h − q) such that (5.38) remains satisfied. Then I 4 = r 2 2 − r 2 1 , and we may take m = 1, 2πp = (h − q).

Concluding Remarks
Surfaces of constant mean curvature or constant Gaussian curvature are other special surfaces that appear amenable to the methods of this paper. A possible alternative general approach, motivated by the analyses of Payne and Weinberger [38] and of Dafermos [11] (see also Horgan and Knowles [21]) for bounded regions in Euclidean space subject, for example, to homogeneous Neumann boundary data, similarly seeks to apply variational arguments to the bounded manifold. In Euclidean space, the Euler-Lagrange equations are the Navier equilibrium equations of linear isotropic homogeneous compressible elasticity and Korn's constant is related to values of Poisson's ratio for which uniqueness fails. Variational arguments applied to the bounded two-dimensional manifold should be expected to similarly lead to non-uniqueness in the equilibrium boundary value problem but for a linear elastic shell. The approach awaits investigation as does extension to bounded manifolds with Neumann boundary conditions, and to manifolds without boundary.