BMO and Elasticity: Korn’s Inequality; Local Uniqueness in Tension

In 1972 Fritz John [30] published a uniqueness theorem for Nonlinear Elasticity that, until recently, was the only result of its kind. He showed that, given a stress-free reference configuration whose elasticity tensor is uniformly positive definite, there is an \(L^{\infty }\)-neighborhood of the reference configuration in the space of strains, rather than the space of deformation gradients, in which there is at most one smooth solution of the equations of equilibrium for the pure-displacement problem for a hyperelastic body.

John’s proof made use of the space of functions of Bounded Mean Oscillation, \(\operatorname{BMO}\), a space that John and Nirenberg [32] had invented some ten years earlier. According to L. Nirenberg ([31, pp. 707–709]), the idea of considering functions whose mean oscillation is bounded was conceived by Fritz John. His motivation appears to have been the analysis of problems in Nonlinear Elasticity, where John had noticed that mappings with small nonlinear strain (see (5.16)) correspond to deformation gradients the are small in \(\operatorname{BMO}\) (see [29] or, e.g., [43, Proposition 4.3 and Lemma 5.6]).

Subsequently, although our understanding of the space \(\operatorname{BMO}\) and its applicability to both Harmonic Analysis and Partial Differential Equations has advanced significantly (see, e.g., the exposition by R. V. Kohn [35, p. 509-512]), the original goal of making use of \(\operatorname{BMO}\) in problems of Elasticity has not progressed. Recently, the authors [43] extended John’s uniqueness result to include the mixed problem. In particular we showed that, given a smooth equilibrium solution \({\mathbf{u}}_{\mathrm{e}}\) at which the second variation of the energy is uniformly positive, there are no other equilibrium solutions \({\mathbf{v}}_{\mathrm{e}}\) for which the difference of the right Cauchy-Green strain tensors: is small in \(L^{\infty }\). Here \(\nabla {\mathbf{u}}\) denotes the deformation gradient: an \(n\) by \(n\) matrix of partial derivatives of the components of the deformation \({\mathbf{u}}:{\varOmega }\to {\mathbb{R}}^{n}\), \((\nabla {\mathbf{u}})^{\mathrm{T}}\) denotes the transpose of \(\nabla {\mathbf{u}}\), and we identify the body with the (bounded) region \({\overline{\varOmega }}\subset {\mathbb{R}}^{n}\) that it occupies in a fixed reference configuration.

In this manuscript we extend the results obtained in [30, 43]. We note that when \({\varOmega }\) has sufficiently smooth boundary (Lipschitz suffices), the space \(\operatorname{BMO}({\varOmega })\) is a Banach space that is between \(L^{\infty }\) and all of the other \(L^{p}\)-spaces, that is, for all \(p\in [1,\infty )\), Specifically, \({[\kern -0.3ex] \cdot [\kern -0.3ex]}_{\operatorname{BMO}({\varOmega })}\le 2 \|\cdot \|_{L^{\infty }({\varOmega })}\) and hence an \(\varepsilon \)-neighborhood in \(\operatorname{BMO}\) is larger than an \(\varepsilon \)-neighborhood in \(L^{\infty }\). Here \({[\kern -0.3ex] \cdot [\kern -0.3ex]}\) denotes the standard seminorm on \(\operatorname{BMO}({\varOmega })\) (see (2.3)).

We show, in particular, that the \(L^{\infty }\)-neighborhood in which there is at most one solution can be enlarged to a neighborhood in \(\operatorname{BMO}\) for both the displacement and the mixed problem provided the equilibrium solution \({{\mathbf{u}}_{\mathrm{e}}}\) has nonnegative principal stresses everywhere. Thus, in this case the strain difference in (1.1) need no longer be uniformly small, but instead it need only be small in the space \(\operatorname{BMO}({\varOmega })\).

There are similar interesting results in the Calculus of Variations literature. Kristensen and Taheri [36, Sect. 6] and Campos Cordero [6, Sect. 4] (see, also, Firoozye [16]) have shown that, for the Dirichlet problem, if \({{\mathbf{u}}_{\mathrm{e}}}\) is a Lipschitz-continuous solution of the equilibrium equations at which the second variation is uniformly positive, then there is a neighborhood of \(\nabla {{\mathbf{u}}_{\mathrm{e}}}\) in \(\operatorname{BMO}\) in which all Lipschitz mappings have energy that is greater than or equal to the energy of \({{\mathbf{u}}_{\mathrm{e}}}\). We note that the assumptions in [6], in particular, are incompatible with the blowup of the energy as the Jacobian goes to zero. Recently [44] we have extended the results in [6, Sect. 4] to include the Neumann and mixed problems. Although our proofs are not applicable to elasticity, we have shown that given a Lipschitz-continuous solution \({{\mathbf{u}}_{\mathrm{e}}}\) of the equilibrium equations at which the second variation is uniformly positive, there is a neighborhood of \(\nabla {{\mathbf{u}}_{\mathrm{e}}}\) in \(\operatorname{BMO}\) in which all mappings \({\mathbf{v}}\) in the Sobolev space \(W^{1,1}({\varOmega };{\mathbb{R}}^{n})\) with \(\nabla {\mathbf{v}}\in \operatorname{BMO}({\varOmega })\) have energy that is strictly greater than the energy of \({{\mathbf{u}}_{\mathrm{e}}}\).

We herein also establish a version of Korn’s inequality for \(\operatorname{BMO}\). It is well-known (see, e.g., [1, 21, 28, 46]) that, for all \(p\in (1,\infty )\), a generalized Korn inequality is valid, that is, there is a constant \(K=K(p)=K(p,n,{\varOmega })\) such that for all \({\mathbf{w}}\in W^{1,p}({\varOmega };{\mathbb{R}}^{n})\) that satisfy a suitable constraint that eliminates infinitesimal rotations (e.g., \({\mathbf{w}}=\mathbf{0}\) on \({\mathscr {D}}\subset \partial {\varOmega }\)). We show that there exists a constant \({\mathscr {K}}={\mathscr {K}}(n)\) such that, for every nonempty, bounded open set \(U\subset {\mathbb{R}}^{n}\), for every \({\mathbf{w}}\in W_{\operatorname{loc}}^{1,1}(U;{\mathbb{R}}^{n})\) with \(\nabla {\mathbf{w}}\in \operatorname{BMO}(U)\). Note that, unlike the standard Korn inequalities, which are only valid for John domains (see [28]) and for which the Korn constant depends on the domain, (1.2) is valid for all bounded open sets \(U\) with a constant that is independent of \(U\). (The lack of a constraint to eliminate infinitesimal rotations is due to the nature of the \(\operatorname{BMO}\)-seminorm. See (2.2) and (3.1).)

Before we present a more detailed description of our results, we note that there is a long history of both nonuniqueness, e.g., buckling [40], and uniqueness results in nonlinear elasticity. Rather than providing details here we instead refer the reader to the introductions of two recent papers concerning uniqueness [41, 43]. These papers also discuss interesting possible extensions of such results: the pure-traction problem, incompressible materials, and live loading, none of which are considered in this manuscript.

We begin in Sect. 2 with our notations. In Sect. 2.1 we then present certain consequences of the Geometric Rigidity theory of Friesecke, James, and Müller [17] (see, also, Conti and Schweizer [11] and Kohn [34]) that are useful in our work. In particular, a result of Lorent [37] as well as a result of Ciarlet and Mardare [10] give conditions under which the equality of two strains, \((\nabla {\mathbf{u}})^{\mathrm{T}}\nabla {\mathbf{u}}\equiv (\nabla { \mathbf{v}})^{\mathrm{T}}\nabla {\mathbf{v}}\), yields the equality of the underlying deformations: \({\mathbf{u}}\equiv {\mathbf{v}}\). (This need not be true without further assumptions, even if \({\mathbf{u}}={\mathbf{v}}\) on \(\partial {\varOmega }\)).

After reviewing certain standard properties of the space \(\operatorname{BMO}\), we then present, in Sect. 2.3, theorems from Harmonic Analysis that we have found useful in this work. Of particular consequence is a result from [43]: If \({\varOmega }\) is a Lipschitz domain and \(1\le p< q<\infty \), then there is a constant \(C=C(p,q,{\varOmega })\) such that, for all \(\psi \in \operatorname{BMO}({\varOmega })\), where \(\langle \psi \rangle _{{\varOmega }}\) denotes the average value of the function \(\psi \) on \({\varOmega }\). This interpolation inequality has a number of important consequences. Specifically, we show that it implies that a result that John and Nirenberg [32] established for cubes is in fact valid for every nonempty, bounded, open region \(V\subset {\mathbb{R}}^{n}\): For all \(q\in (1,\infty )\) there exists a constant \(C=C(q)\) such that for all \(\phi \in \operatorname{BMO}(V)\), where the supremum is taken over all cubes \(Q\) that are compactly supported in \(V\) and have faces that are parallel to the coordinate planes. (If \(q=1\) the quantity in the center of inequality (1.4) is equal to the \(\operatorname{BMO}\)-seminorm of \(\phi \).) In Sect. 3 we then make use of (1.4) together with a version of Korn’s inequality due to Diening, Růžička, and Schumacher [13] to establish Korn’s inequality in \(\operatorname{BMO}\), that is, (1.2).

In Sect. 4 we introduce our hypotheses on a compressible, nonlinearly hyperelastic body where the stored-energy density \(\sigma \) depends on the material point \({\mathbf{x}}\) and the right Cauchy-Green strain tensor \({\mathbf{C}}_{{\mathbf{u}}}({\mathbf{x}})=[\nabla {\mathbf{u}}({ \mathbf{x}})]^{\mathrm{T}}\nabla {\mathbf{u}}({\mathbf{x}})\). Thus, in the absence of body forces and surface tractions, the total energy of a deformation \({\mathbf{u}}:\overline{\varOmega }\to {\mathbb{R}}^{n}\), which satisfies \({\mathbf{u}}={\mathbf{d}}\) on \({\mathscr {D}}\subset \partial {\varOmega }\), is given by

The second variation of ℰ evaluated at a solution of the corresponding equilibrium equations \({{\mathbf{u}}_{\mathrm{e}}}\) is then equal to where \({\mathbb{C}}=4\frac{\partial ^{2}}{\partial {\mathbf{C}}^{2}}\sigma ({ \mathbf{x}},{\mathbf{C}})\) denotes the elasticity tensor, \({\mathbf{K}}=2\frac{\partial }{\partial {\mathbf{C}}}\sigma ({\mathbf{x}},{ \mathbf{C}})\) denotes the (second) Piola-Kirchhoff stress tensor, \({\mathbf{E}}=(\nabla {{\mathbf{u}}_{\mathrm{e}}})^{\mathrm{T}}\nabla { \mathbf{w}}+(\nabla {\mathbf{w}})^{\mathrm{T}}\nabla {{\mathbf{u}}_{ \mathrm{e}}}\), and \({\mathbf{w}}\in W^{1,2}({\varOmega };{\mathbb{R}}^{n})\) satisfies \({\mathbf{w}}=\mathbf{0}\) on \({\mathscr {D}}\subset \partial {\varOmega }\).

If \({\mathbf{K}}\) is positive semi-definite, equivalently, the principal stresses are nonnegative, and ℂ is uniformly positive definite, then \(\delta ^{2}{\mathscr {E}}({{\mathbf{u}}_{\mathrm{e}}})\) is uniformly positive. Standard techniques (see the introduction to [43] with particular attention to equations (1.3) and (1.4) on p. 411), which are usually applied in the space of deformation gradients, make use of Taylor’s theorem to deduce that there is then an \(L^{\infty }\) neighborhood of \({\mathbf{C}}_{{\mathbf{u}}_{\mathrm{e}}}\) in strain space (see (1.1)) in which there are no other solutions of the equilibrium equations. A refinement of this argument, which is due to John [30, pp. 624–625] (again, see the introduction to [43] with particular attention to equation (1.5) on p. 412), makes use of (1.3) to enlarge the set in which there are no other solutions to a neighborhood of \({\mathbf{C}}_{{\mathbf{u}}_{\mathrm{e}}}\) in the space \(\operatorname{BMO}\). We present the details of this argument in Sect. 5 of this manuscript. We also note, in Sect. 5.1, how these results simplify when one of the two right Cauchy-Green strain tensors is in an \(L^{\infty }\)-neighborhood of the reference configuration. Finally, in Sect. 6, we present further simplifications that occur when the reference configuration is itself at equilibrium.

For any domain (nonempty, connected, open set) \(U\subset {\mathbb{R}}^{n}\), \(n\ge 2\), we denote by \(L^{p}(U)\), \(p\in [1,\infty )\), the space of real-valued Lebesgue measurable functions \(\psi \) whose \(L^{p}\)-norm is finite: \(L^{\infty }(U)\) will denote those Lebesgue measurable functions whose essential supremum is finite. \(L^{1}_{\operatorname{loc}}(U)\) will consist of those Lebesgue measurable functions that are integrable on every compact subset of \(U\). We shall write \(C(U;{\mathbb{R}}^{n})\) for the set of continuous functions \({\mathbf{u}}:U\to {\mathbb{R}}^{n}\), while \(C^{1}(\overline{U};{\mathbb{R}}^{n})\) will denote those continuous functions \({\mathbf{u}}:\overline{U}\to {\mathbb{R}}^{n}\) whose classical derivative exists on \(U\) and has an extension that is continuous on \(\overline{U}\), where \(\overline{U}\) denotes the closure of \(U\).

We shall write \({\varOmega }\subset {\mathbb{R}}^{n}\), \(n\ge 2\), to denote a Lipschitz domain, that is, a bounded domain whose boundary \(\partial {\varOmega }\) is (strongly) Lipschitz. (See, e.g., [14, p. 127], [39, p. 72], or [26, Definition 2.5].) Essentially, a bounded domain is Lipschitz if, in a neighborhood of every boundary point, the boundary is the graph of a Lipschitz-continuous function and the domain is on “one side” of this graph.

For \(1\le p\le \infty \), \(W^{1,p}(\varOmega ;{\mathbb{R}}^{N})\) will denote the usual Sobolev space of (Lebesgue) measurable (vector-valued) functions \({\mathbf{u}}\in L^{p}(\varOmega ;{\mathbb{R}}^{N})\) whose distributional gradient \(\nabla {\mathbf{u}}\) is also contained in \(L^{p}\). If \(\phi \in W^{1,p}(\varOmega ):=W^{1,p}(\varOmega ;{\mathbb{R}})\) we shall denote its \(W^{1,p}\)-norm by¹ We shall write \(W^{1,p}_{0}(\varOmega ;{\mathbb{R}}^{N})\) for the subspace of \({\mathbf{u}}\in W^{1,p}(\varOmega ;{\mathbb{R}}^{N})\) that satisfy \({\mathbf{u}}=\mathbf{0}\) on \(\partial \varOmega \) (in the sense of trace). \(W_{\operatorname{loc}}^{1,p}(U;{\mathbb{R}}^{N})\) will denote the set of \({\mathbf{u}}\in W^{1,p}(V;{\mathbb{R}}^{N})\) for every domain \(V{\subset \!\subset }U\), where we write \(V{\subset \!\subset }U\) provided that \(V\subset K_{V} \subset U\) for some compact set \(K_{V}\).

We shall write \({\mathbb{M}}^{n\times n}\) for the (vector) space of \(n\) by \(n\) matrices with real entries. Given an orthonormal basis \({\mathbf{e}}_{i}\), \(i=1,2,\dots ,n\), for \({\mathbb{R}}^{n}\) we write \(a_{i}={\mathbf{a}}\cdot {\mathbf{e}}_{i}\) for \({\mathbf{a}}\in {\mathbb{R}}^{n}\) and \(F_{ij}={\mathbf{e}}_{i}\cdot {\mathbf{F}}{\mathbf{e}}_{j}\) for \({\mathbf{F}}\in {\mathbb{M}}^{n\times n}\). The set of symmetric and positive-definite symmetric matrices in \({\mathbb{M}}^{n\times n}\) shall be denoted by respectively, where \({\mathbf{H}}^{\mathrm{T}}\) denotes the transpose of \({\mathbf{H}}\in {\mathbb{M}}^{n\times n}\). We write \({\mathbf{H}}:{\mathbf{K}}:=\operatorname{trace}({\mathbf{H}}{\mathbf{K}}^{\mathrm{T}})\) for the inner product of \({\mathbf{H}},{\mathbf{K}}\in {\mathbb{M}}^{n\times n}\). The norm of \({\mathbf{H}}\in {\mathbb{M}}^{n\times n}\) is then given by \(|{\mathbf{H}}|:=\sqrt{{\mathbf{H}}:{\mathbf{H}}\,}\). We write for the group of rotations, where \({\mathbf{I}}\) denotes the identity matrix and \(\det {\mathbf{F}}\) denotes the determinant of \({\mathbf{F}}\in {\mathbb{M}}^{n\times n}\).

In this section we obtain a version of Korn’s inequality that involves the \(\operatorname{BMO}\)-seminorm of both the gradient of a function and the symmetric part of its gradient. Our result is a simple consequence of the following result of Diening, Růžička, and Schumacher.

In the remainder of this manuscript we shall focus on the minimization problem that arises when one considers the theory of Nonlinear (Finite) Hyperelasticity.

The first result of this section yields a comparison of the energy of an equilibrium solution, \({{\mathbf{u}}_{\mathrm{e}}}\), to the energy of any admissible deformation whose strains are sufficiently close to \({\mathbf{C}}_{\mathrm{e}}\) in \(\operatorname{BMO}\cap \,L^{1}\). Our main theorem then follows from this energy estimate. It establishes that, given a solution \({{\mathbf{u}}_{\mathrm{e}}}\) of the equilibrium equations whose principal stresses are positive (or a smooth solution whose principal stresses are sufficiently small and negative) and where the integral of the elasticity tensor is uniformly positive, there is a neighborhood of \({\mathbf{C}}_{\mathrm{e}}\) in \(\operatorname{Psym_{\mathit{n}}}\) in the \(\operatorname{BMO}\cap \,L^{1}\)-topology in which there are no other solutions of the equilibrium equations.

We here note that the statement of Theorem 5.2 simplifies when the body in its reference configuration is itself at equilibrium. Thus, we assume that \({{\mathbf{u}}_{\mathrm{e}}}={\mathbf{id}}\in C^{1}({ \overline{\varOmega }};{\mathbb{R}}^{n})\), i.e., where \({\varOmega }\subset {\mathbb{R}}^{n}\) is a Lipschitz domain. Clearly, we also require that \({\mathbf{d}}={\mathbf{id}}\) on \({\mathscr {D}}\). However, we do not require that this reference configuration be stress free.