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We establish a family of coercive Korn-type inequalities for generalized incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the existence theory for a multitude of models in continuum mechanics in an optimal way. Different from our preceding work [F. Gmeineder, P. Lewintan and P. Neff, Optimal incompatible Korn–Maxwell–Sobolev inequalities in all dimensions, Calc. Var. PDE 62 (2023) 182], where we focused on the case p = 1 and incompatibilities governed by the matrix curl, the case p > 1 considered in this paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalizations of recently proved inequalities by P. Lewintan, S. Müller and P. Neff [Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy, Calc. Var. PDE 60 (2021) 150] in the realm of incompatible Korn-type inequalities with conformally invariant dislocation energy. However, being applicable to higher-order scenarios as well, our approach equally gives the first and sharp inequalities involving Kröner's incompability tensor inc. [ABSTRACT FROM AUTHOR]

Korn's inequality has been at the heart of much exciting research since its first appearance in the beginning of the 20th century. Many are the applications of this inequality to the analysis and construction of discretizations of a large variety of problems in continuum mechanics. In this paper, we prove that the classical Korn inequality holds true in anisotropic Sobolev spaces. We also prove that an extension of Korn's inequality, involving non-linear continuous maps, is valid in such spaces. Finally, we point out that another classical inequality, namely Poincaré's inequality, also holds in anisotropic Sobolev spaces. [ABSTRACT FROM AUTHOR]

Korn's first inequality states that there exists a constant such that the L 2 -norm of the infinitesimal displacement gradient is bounded above by this constant times the L 2 -norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body ℬ. This inequality is known to hold when the L 2 -norm is replaced by the L p -norm for any p ∈ (1 , ∞) . However, if p = 1 or p = ∞ the resulting inequality is false. It was previously shown that if one replaces the L ∞ -norm by the BMO -seminorm (Bounded Mean Oscillation) then one maintains Korn's inequality. (Recall that L ∞ (B) ⊂ BMO (B) ⊂ L p (B) ⊂ L 1 (B) , 1 < p < ∞ .) In this manuscript it is shown that Korn's inequality is also maintained if one replaces the L 1 -norm by the norm in the Hardy space H 1 , the predual of BMO . One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of ℬ. [ABSTRACT FROM AUTHOR]

In the first part of this article, we present a new proof for Korn’s inequality in an n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude.

In this paper, we study certain inequalities and a related result for weighted Sobolev spaces on Hölder-α domains, where the weights are powers of the distance to the boundary. We obtain results regarding the divergence equation's solvability, and the improved Poincaré, the fractional Poincaré, and the Korn inequalities. The proofs are based on a local-to-global argument that involves a kind of atomic decomposition of functions and the validity of a weighted discrete Hardy-type inequality on trees. The novelty of our approach lies in the use of this weighted discrete Hardy inequality and a sufficient condition that allows us to study the weights of our interest. As a consequence, the assumptions on the weight exponents that appear in our results are weaker than those in the literature. [ABSTRACT FROM AUTHOR]

We characterise all linear maps A : R n × n → R n × n such that, for 1 ≤ p < n , P L p ∗ (R n) ≤ c ( A [ P ] L p ∗ (R n) + Curl P L p (R n) ) holds for all compactly supported P ∈ C c ∞ (R n ; R n × n) , where Curl P displays the matrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different combinations between the ellipticities of A , the integrability p and the underlying space dimensions n, especially requiring a finer analysis in the two-dimensional situation. [ABSTRACT FROM AUTHOR]

In the first part of this article, we present a new proof for Korn's inequality in an n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude. [ABSTRACT FROM AUTHOR]

It is known that the famous theoretical formula by Koiter for the critical buckling load of circular cylindrical shells under axial compression does not coincide with the experimental data. Namely, while Koiter’s formula predicts linear dependence of the buckling load λ (h) of the shell thickness h ( h > 0 is a small parameter), one observes the dependence λ (h) ∼ h 3 / 2 in experiments; i.e., the shell buckles at much smaller loads for small thickness. This theoretical prediction failure is believed to be caused by the so-called sensitivity to imperfections phenomenon (both, shape and load). Grabovsky and the first author have rigorously proven in (J Nonlinear Sci 26(1):83–119, 2016) , that in the problem of circular cylindrical shells buckling under axial compression, a small load twist leads to the buckling load scaling λ (h) ∼ h 5 / 4 , while shape imperfections are likely to result in the scaling λ (h) ∼ h 3 / 2. In this work, we prove that in fact the buckling load λ (h) of cylindrical (not necessarily circular) shells under vertical compression depends on the curvature of the cross section curve. When the cross section is a convex curve with uniformly positive curvature, then λ (h) ∼ h , and when the cross-sectional curve has positive curvature except at finitely many points, then C 1 h 8 / 5 ≤ λ (h) ≤ C 2 h 3 / 2 for h small thickness h > 0. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. The first goal is to provide a simple and constructive proof of Korn inequalities in half-space with weighted norms. The proof leads to explicit values of the constants. The second objective is to use these inequalities to show that the linear elasticity system in half-space admits a coercive variational formulation. This formulation corresponds to the physical case in which the solution is evanescent at infinity.

We consider a viscous fluid in a finite-depth domain of two dimensions, with a free moving boundary and a fixed solid boundary. The fluid dynamics are governed by gravity-driven incompressible micropolar equations, and the surface tension is neglected on the free surface. The main result of this paper is to prove the global well-posedness of the surface wave problem when the fluid domain is horizontally periodic. And the solutions decay to equilibrium at an almost exponential rate. This is achieved by establishing delicate estimates when dealing with the strong coupling of fluid velocity and micro-rotation velocity. Our proof relies on two-tier nonlinear energy method developed by Guo and Tice [17–19]. [ABSTRACT FROM AUTHOR]

The general theory of slender structure buckling by Grabovsky and Truskinovsky (Cont. Mech. Thermodyn. 19(3–4):211-243, 2007), (later extended in J. Nonlinear Sci. 26(1):83–119, 2016 by Grabovsky and the author), predicts that the critical buckling load of a thin shell under dead loading is closely related to the Korn's constant (in Korn's first inequality) of the shell under the Dirichlet boundary conditions resulting from the loading program. It is known that under zero Dirichlet boundary conditions on the thin part of the boundary of positive, negative, and zero (one principal curvature vanishing, and one apart from zero) Gaussian curvature shells, the optimal Korn constant in Korn's first inequality scales like the thickness to the power of −1, − 4 / 3 , and − 3 / 2 respectively. In this work we analyse the scaling of the optimal constant in Korn's first inequality for elliptic shells that contain a finite number of points where both principal curvatures vanish. We prove that the presence of at least one such point on the shell leads to the scaling drop from the thickness to the power of −1 to the thickness to the power of − 3 / 2 . To our best knowledge, this is the first result in the direction for constant-sign curvature shells, that do not contain a developable region. In addition, under the assumption that a suitable trivial branch exists, we prove that in fact the buckling deformation of such shells under dead loading, should be localized at the vanishing curvature points, as the shell thickness h goes to zero. [ABSTRACT FROM AUTHOR]

We consider a smooth, compact and embedded hypersurface Σ without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded H ∞ -calculus with angle smaller than π / 2 . As an application, we consider critical spaces for the Navier–Stokes equations on the surface Σ . In case Σ is two-dimensional, we show that any solution with a divergence-free initial value in L 2 (Σ , T Σ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field. [ABSTRACT FROM AUTHOR]

In this paper, we show that the so-called Korn inequality holds for vector fields with a zero normal or tangential trace on a subset (of positive measure) of the boundary of Lipschitz domains. We further show that the validity of this inequality depends on the geometry of this subset of the boundary. We then consider three eigenvalue problems for the Lamé operator: we constrain the traction in the tangential direction and the normal component of the displacement, the related problem of constraining the normal component of the traction and the tangential component of the displacement, and a third eigenproblem that considers mixed boundary conditions. We show that eigenpairs for these eigenproblems exist on a broad variety of domains. Analytic solutions for some of these eigenproblems are given on simple domains. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. The first goal is to provide a simple and constructive proof of Korn inequalities in half-space with weighted norms. The proof leads to explicit values of the constants. The second objective is to use these inequalities to show that the linear elasticity system in half-space admits a coercive variational formulation. This formulation corresponds to the physical case in which the solution is evanescent at infinity. [ABSTRACT FROM AUTHOR]

For 1 < p < ∞, we prove an Lp‐version of the generalized Korn inequality for incompatible tensor fields P in W01,p(Curl;Ω,ℝ3×3). More precisely, let Ω⊂ℝ3 be a bounded Lipschitz domain. Then there exists a constant c = c(p, Ω) > 0 such that ‖P‖Lp(Ω,ℝ3×3)≤c‖symP‖Lp(Ω,ℝ3×3)+‖CurlP‖Lp(Ω,ℝ3×3)holds for all tensor fields P∈W01,p(Curl;Ω,ℝ3×3), that is, for all P∈W1,p(Curl;Ω,ℝ3×3) with vanishing tangential trace P×ν=0 on ∂Ω where ν denotes the outward unit normal vector field to ∂Ω. For compatible P=Du, this recovers an Lp‐version of the classical Korn's first inequality and for skew‐symmetric P=A an Lp‐version of the Poincaré inequality. [ABSTRACT FROM AUTHOR]

In this note, we study the complex constant rank condition for differential operators and its implications for coercive differential inequalities. These are inequalities of the form ‖ A u ‖ L p ≤ ‖ A u ‖ L q , for exponents 1 ≤ p , q < ∞ and homogeneous constant-coefficient differential operators A and A. The functions u : Ω → R d are defined on open and bounded sets Ω ⊂ R N satisfying certain regularity assumptions. Depending on the order of A and A , such an inequality might be viewed as a generalisation of either Korn's or Sobolev's inequality, respectively. In both cases, as we are on bounded domains, we assume that the Fourier symbol of A satisfies an algebraic condition, the complex constant rank property. [ABSTRACT FROM AUTHOR]

We consider the scaling of the optimal constant in Korn's first inequality for elliptic and parabolic shells which was first given by Grabovsky and Harutyunyan with hints coming from the test functions constructed by Tovstik and Smirnov on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell is given by one single principal coordinate, in particularly, including closed elliptic shells. [ABSTRACT FROM AUTHOR]

We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler 's foundational work [64] , symmetric quasiconvexity is the pivotal notion as regards sequential weak*-lower semicontinuity and hence for the existence of minima of the relaxed functionals on BD. The overarching main difficulty here is the lack of Korn 's Inequality in the L 1 -setting, hereby implying that the BD-case is genuinely different from the study of variational integrals on BV. Unlike for superlinear growth, symmetric quasiconvex functionals, where we establish partial regularity by direct reduction to the full gradient case by Korn -type inequalities, such a reduction does not work in the linear growth case and identifies the latter as the only situation requiring a treatment on its own. [ABSTRACT FROM AUTHOR]

In this manuscript two BMO estimates are obtained, one for Linear Elasticity and one for Nonlinear Elasticity. It is first shown that the BMO-seminorm of the gradient of a vector-valued mapping is bounded above by a constant times the BMO-seminorm of the symmetric part of its gradient, that is, a Korn inequality in BMO. The uniqueness of equilibrium for a finite deformation whose principal stresses are everywhere nonnegative is then considered. It is shown that when the second variation of the energy, when considered as a function of the strain, is uniformly positive definite at such an equilibrium solution, then there is a BMO-neighborhood in strain space where there are no other equilibrium solutions. [ABSTRACT FROM AUTHOR]

This work is part of a program of development of asymptotically sharp geometric rigidity estimates for thin domains. A thin domain in three dimensional Euclidean space is roughly a small neighborhood of regular enough two dimensional compact surface. We prove an asymptotically sharp geometric rigidity interpolation inequality for thin domains with little regularity. In contrast to that celebrated Friesecke et al. rigidity estimate [Commun. Pure Appl. Math. 55(11):1461–1506, 2002] for plates, our estimate holds for any proper rotations R ∈ S O (3) . Namely, the estimate bounds the L p distance of the gradient of any y ∈ W 1 , p (Ω , R 3) field from any constant proper rotation R ∈ S O (3) , in terms of the average L p distance (nonlinear strain) of the gradient ∇ y from the rotation group S O (3) , and the average L p distance of the field itself from the set of rigid motions corresponding to the rotation R . There are several remarkable facts about the estimate: 1. The constants in the estimate are sharp in terms of the domain thickness scaling for any thin domains with the required regularity. 2. In the special cases when the domain has positive or negative Gaussian curvature, the inequality reduces the problem of estimating the gradient ∇ y in terms of the prototypical nonlinear strain ∫ Ω dist p (∇ y (x) , S O (3)) d x to the easier problem of estimating only the vector field y in terms of the nonlinear strain without any loss in the constant scalings as the Ansätze suggest. The later will be a geometric rigidity Korn-Poincaré type estimate. This passage is major progress in the thin domain rigidity problem. 3. For the borderline energy scaling (bending-to-stretching), the estimate implies improved strong compactness on the vector fields for free. Finally, this being said, our new interpolation inequality reduces the problem of proving "any" geometric one well rigidity problem in thin domains to estimating the vector field itself instead of the gradient, thus reducing the complexity of the problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a reduced Crouzeix–Raviart immersed finite element method (RCRIFEM) for two-dimensional elasticity problems with interface, which is based on the Kouhia–Stenberg finite element method (Kouhia et al. 1995) and Crouzeix–Raviart IFEM (CRIFEM) (Kwak et al. 2017). We use a P 1 {P_{1}} -conforming like element for one of the components of the displacement vector, and a P 1 {P_{1}} -nonconforming like element for the other component. The number of degrees of freedom of our scheme is reduced to two thirds of CRIFEM. Furthermore, we can choose penalty parameters independent of the Poisson ratio. One of the penalty parameters depends on Lamé's second constant μ, and the other penalty parameter is independent of both μ and λ. We prove the optimal order error estimates in piecewise H 1 {H^{1}} -norm, which is independent of the Poisson ratio. Numerical experiments show optimal order of convergence both in L 2 {L^{2}} and piecewise H 1 {H^{1}} -norms for all problems including nearly incompressible cases. [ABSTRACT FROM AUTHOR]

In this work, we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients. [ABSTRACT FROM AUTHOR]

We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to prove a fractional version of the classical Korn's inequality. [ABSTRACT FROM AUTHOR]

In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method of Di Pietro and Ern (2015), that requires the use of polynomials of degree k ≥ 1 for stability. Specifically, we show that coercivity can be recovered for k = 0 by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces. This term plays a key role in the fulfilment of a discrete Korn inequality on broken polynomial spaces, for which a novel proof valid for general polyhedral meshes is provided. Locking-free error estimates are derived for both the energy- and the L 2 -norms of the error, that are shown to convergence, for smooth solutions, as h and h 2 , respectively (here, h denotes the meshsize). A thorough numerical validation on a complete panel of two- and three-dimensional test cases is provided. [ABSTRACT FROM AUTHOR]

In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng , 124 , 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf–sup condition hold uniformly in the mesh size and also for a nonempty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia–Stenberg FEM to three space dimensions: the finite element space with one nonconforming and two conforming piecewise affine components does not satisfy a discrete inf–sup condition with piecewise constant pressure approximations, while finite element functions with two nonconforming and one conforming component do not satisfy a discrete Korn inequality. [ABSTRACT FROM AUTHOR]

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability. [ABSTRACT FROM AUTHOR]

This paper studies possibilities to use Neumann's method to solve boundary problems of elastic thin shells. Variational statement of statical problems for shells allows examining the problems within the space of distributions. Convergence of the Neumann's method is proved for the shells with holes when the boundary of the domain is not completely fixed. Numerical implementation of the Neumann's method normally takes a lot of time before some reliable results can be achieved. This paper suggests a way to improve convergence of the process and allows for parallel computing and checkout procedure during calculations.

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain Ω ⊂ R² with C1-boundary there is a corresponding partition Ω = Ω1 ⋃ ... ⋃ ΩN with Σj=1NH¹(∂Ωj\∂Ω)≤θ such that each component is a John domain with a John constant only depending on θ. The result implies that many inequalities in Sobolev spaces such as Poincaré's or Korn's inequality hold on the partition of Ω for uniform constants, which are independent of Ω. [ABSTRACT FROM AUTHOR]

We present a piecewise Korn inequality for generalized special functions of bounded deformation (GSBD2) in a planar setting generalizing the classical result in elasticity theory to the setting of functions with jump discontinuities. We show that for every configuration there is a partition of the domain such that on each component of the cracked body the distance of the function from an infinitesimal rigid motion can be controlled solely in terms of the linear elastic strain. In particular, the result implies that GSBD2 functions have bounded variation after subtraction of a piecewise infinitesimal rigid motion. As an application we prove a density result in GSBD2 in dimension two. Moreover, for all d ≥ 2 we show GSBD2 (Ω) ⊂ (GBV (Ω; R))d and the embedding SBD2 (Ω) ∩ L∞(Ω; Rd) ,→ SBV (Ω; Rd) into the space of special functions of bounded variation (SBV ). Finally, we present a Korn-Poincaré inequality for functions with small jump sets in arbitrary space dimension. [ABSTRACT FROM AUTHOR]

We discuss the value of the best constant in Gaffney inequality namely ‖ ∇ ω ‖ L 2 2 ≤ C ( ‖ d ω ‖ L 2 2 + ‖ δ ω ‖ L 2 2 + ‖ ω ‖ L 2 2 ) when either ν ∧ ω = 0 or ν ⌟ ω = 0 on ∂Ω. [ABSTRACT FROM AUTHOR]

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections. [ABSTRACT FROM AUTHOR]

We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in with a sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly, the result shows that each function in has bounded variation away from an arbitrarily small part of the domain. [ABSTRACT FROM AUTHOR]

We propose a 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations. The low-order conforming finite elements are not inf-sup stable. The low-order nonconforming finite elements do not satisfy the Korn inequality. The proposed finite element space consists of two conforming components and one nonconforming component. We prove that the discrete inf-sup condition is valid and the discrete Korn inequality holds uniformly in the mesh-size. Based on these results we give some numerical verification. In addition, this element is compared numerically with six other mixed finite elements. [ABSTRACT FROM AUTHOR]

The possibility of using Neumann's method to solve the boundary problems for thin elastic shells is studied. The variational statement of the static problems for the shells allows for a problem examination within the distribution space. The convergence of Neumann's method is proven for the shells with holes when the boundary of the domain is not completely fixed. The numerical implementation of Neumann's method normally requires significant time before any reliable results can be achieved. This paper suggests a way to improve the convergence of the process, and allows for parallel computing and evaluation during the calculations. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the nonsymmetric version of the Nitsche method without penalty for compressible and incompressible elasticity. For the compressible case we prove convergence of the error in the H¹- and L²-norms. In the incompressible case we use a Galerkin least squares pressure stabilization and we prove convergence in the H¹-norm for the velocity and convergence of the pressure in the L²-norm. [ABSTRACT FROM AUTHOR]

We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity. [ABSTRACT FROM AUTHOR]