Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient

In this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values. Geometrically, this can be interpreted as finding a minimal Lipschitz extension. Due to the weak convexity of the functional associated to this problem, solutions are generally nonunique. By adopting G. Aronsson's notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation. In fact, we actually prove a weak maximum principle for this partial differential equation, which in some sense is the Euler equation for the minimization problem. This is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity. To overcome this difficulty we use the weak solutions of M. G. Crandall and P.-L. Lions, also known as viscosity solutions, in conjunction with some arguments using integration by parts.     

On Strict Convexity and Continuous Differentiability of Potential Functions in Optimal Transportation

This note concerns the relationship between conditions on cost functions and domains, the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings. In particular, we prove that if the cost function satisfies the condition (A3), introduced in our previous work with Xinan Ma, the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function, then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function. Our results extend, by different and more direct proof, similar results of Loeper proved by approximation from our earlier work on regularity.     

On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations

The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford, New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The stored energy function of such materials has the form

Convexity and symmetrization in relativistic theories

There is a strong motivation for the desire to have symmetric hyperbolic field equations in thermodynamics, because they guarantee well-posedness of Cauchy problems. A generic quasi-linear first order system of balance laws — in the non-relativistic case — can be shown to be symmetric hyperbolic, if the entropy density is concave with respect to the variables. In relativistic thermodynamics this is not so.This paper shows that there exists a scalar quantity in relativistic thermodynamics whose concavity guarantees a symmetric hyperbolic system. But that quantity — we call it — — is not the entropy, although it is closely related to it. It is formed by contracting the entropy flux vector — h^a with a privileged time-like congruence .It is also shown that the convexity of h plus the requirement that all speeds be smaller than the speed of light c provide symmetric hyperbolic field equations for all choices of the direction of time.At this level of generality the physical meaning of —h is unknown. However, in many circumstances it is equal to the entropy. This is so, of course, in the non-relativistic limit but also in the non-dissipative relativistic fluid and even in relativistic extended thermodynamics for a non-degenerate gas.     

Canonical and Anti-Canonical Transformations Preserving Convexity of Potentials

The aim of the paper is to characterize transformations that preserve the potential structure of a relationship between dual variables. The first step consists in deriving a geometric definition of the condition for the existence of a potential. Having at hand this formulation, it becomes clear that the canonical similitudes represents the class of transformations that preserves the potential form of a relationship. Next, we derive the conditions under which canonical similitudes preserve the convexity of the potential or change it into concavity. This new class of transformations can be viewed as a generalization of the Legendre-Fenchel transformation. These concepts are applied to the Hooke constitutive relationship.     

Displacement Convexity and Minimal Fronts at Phase Boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition are strictly convex in the sense of displacement convexity under a natural change of variables.We use this to show that, in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals and, in the multi-component setting, jointly displacement convex functionals.     

Convexity of Stokes Waves of Extreme Form

Existence is established of a piecewise-convex, periodic, planar curve S below which is defined a harmonic function which simultaneously satisfies prescribed Dirichlet and Neumann boundary conditions on S. In hydrodynamics this corresponds to the existence of a periodic Stokes wave of extreme form which has a convex profile between consecutive stagnation points where there is a corner with a contained angle of 120°.     

Partial Uniqueness and Obstruction to Uniqueness in Inverse Problems for Anisotropic Elastic Media

We consider the inverse problem of identifying the density and elastic moduli for three-dimensional anisotropic elastic bodies, given displacement and traction measurements made at their surface. These surface measurements are modelled by the dynamic Dirichlet-to-Neumann map on a finite time interval. For linear or nonlinear anisotropic hyperelastic bodies we show that the displacement-to-traction surface measurements do not change when the density and elasticity tensor in the interior are transformed tensorially by a change of coordinates fixing the surface of the body to first order. Our main tool, a new approach in inverse problems for elastic media, is the representation of the equations of motion in a covariant form (following Marsden and Hughes, 1983) that preserves the underlying physics.In the case of classical linear elastodynamics we then investigate how the type of anisotropy changes under coordinate transformations. That is, we analyze the orbits of general linear, anisotropic elasticity tensors under the action by pull-back of diffeomorphisms that fix the surface of the elastic body to first order, and derive a pointwise characterization of parts of the orbits under this action. For example, we show that the orbit of isotropic elastic media, at any point in the body, consists of some transversely isotropic and some orthotropic elastic media. We then derive the first uniqueness result in the inverse problem for anisotropic media using surface displacement-traction data: uniqueness of three elastic moduli for tensors in the orbit of isotropic elasticity tensors. ^†Partially supported by an MSRI Postdoctoral Fellowship. Research at MSRI is supported in part by NSF grant DMS-9850361. This work was conducted while the first author was a Gibbs Instructor at Yale University. ^‡Partially supported by an MSRI Postdoctoral Fellowship, and by NSF grant DMS-9801664 (9996350).     

Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions

We consider a system of N balance laws compatible with an entropy principle and convex entropy density. Using the special symmetric form induced by the main field, we define the concept of principal subsystem associated with the system. We prove that the 2 ^N −2 principal subsystems are also symmetric hyperbolic and satisfy a subentropy law. Moreover we can verify that for each principal subsystem the maximum (minimum) characteristic velocity is not larger (smaller) than the maximum (minimum) characteristic velocity of the full system. These are the subcharacteristic conditions. We present some simple examples in the case of the Euler fluid. Then in the case of dissipative hyperbolic systems we consider an equilibrium principal subsystem and we discuss the consequences in the setting of extended thermodynamics. Finally in the moments approach to the Boltzmann equation we prove, as a consequence of the previous result, that the maximum characteristic velocity evaluated at the equilibrium state does not decrease when the number of moments increases. (Accepted October 6, 1995)     

Stress Distribution at the Contact of Absolutely Rigid Stamp and Elastic Orthotropic Strip

A plane elastic problem for an orthotropic infinite strip with mixed boundary conditions is investigated. A model of the strip has been built by using the method of integral Fourier transforms. We obtain relationships which allow us to formulate singular integral equations for the various types of boundary conditions on one of its edges. The problems in the case of both smooth stamp-strip contact and rigid stamp-strip adhesion have been considered. It is shown that the effect of anisotropy on the contact stress distribution is minor. The stress intensity factors at the stamp corners, which are the main parameters of fracture, are evaluated. The quasi-invariance of a certain combination of the stress intensity factors is confirmed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Uniqueness and Complementary Energy in Nonlinear Elastostatics

Global uniqueness of the smooth stress and deformation to within the usual rigid-body translation and rotation is established in the null traction boundary value problem of nonlinear homogeneous elasticity on a n-dimensional star-shaped region. A complementary energy is postulated to be a function of the Biot stress and to be para-convex and rank-(n-1) convex, conditions analogous to quasi-convexity and rank-(n-2) of the stored energy function. Uniqueness follows immediately from an identity involving the complementary energy and the Piola-Kirchhoff stress. The interrelationship is discussed between the two conditions imposed on the complementary energy, and between these conditions and those known for uniqueness in the linear elastic traction boundary value problem.     

Equivalence between Rank-One Convexity and Polyconvexity for Some Classes of Elastic Materials

Let W(F) = φ(λ ₁ ^s + λ ₂ ^s + λ ₃ ^s ) + ψ(λ ₁ ^r λ ₂ ^r + λ ₁ ^r λ ₃ ^r + λ ₂ ^r λ ₃ ^r ) + f(λ ₁ λ ₂ λ ₃) be a stored energy function. We prove that, for this function, rank-one convexity is equivalent to polyconvexity.under suitable assumptions on φ, ψ and f.     

Uniqueness of uniaxial elastic-plastic interface motions

The uniaxial motion of interfaces between regions deforming elastically and regions deforming plastically is considered. The governing constitutive, stress-rate/strain-rate equations in both elastic and plastic regions are taken to be non-linear. Discontinuity relations across such interfaces are established by the repeated differentiation of existing relations. The relations given by previous workers (especially R.J. Clifton, T.C.T. Ting, E.H. Lee and Th. von Kármán) are discussed. The precise situations in which they hold are considered, and it is shown that some of these relations, while apparently derived for different situations, can, in certain circumstances, be shown to be equivalent. It has been shown that six essentially different types of motion can occur, and, when the constitutive equations are linear, each type of motion is unique. This result is extended to the non-linear situation, by means of an established local expansion procedure. For the case of a meeting interaction of stress waves carrying initially linear profiles, the previous (linear) analysis given by L.W. Morland and A.D. Cox fails to distinguish between certain types of motion. This motion is reconsidered and it is shown how non-linearity in the constitutive laws serves to determine uniquely the type of motion that takes place.     

Uniqueness of free-boundary flows under gravity

A class of steady potential flows of an ideal fluid is considered in which the fluid flows between fixed boundaries and then emerges as a jet with one free boundary. Gravity acts on the fluid perpendicularly to the direction of the jet at infinity downstream. An inverse Froude number α is defined in terms of the flux Q and the depth d of the fluid at the separation point. It is proved that for each α>0 there is at most one flow which reaches to a supercritical uniform stream depth at infinity downstream. Monotonicity properties are proved for various flow parameters, and the behaviour of the flow as α → 0 is described.     

Attractors of Parabolic Equations Without Uniqueness

In this paper we study the existence of global compact attractors for nonlinear parabolic equations of the reaction-diffusion type and variational inequalities. The studied equations are generated by a difference of subdifferential maps and are not assumed to have a unique solution for each initial state. Applications are given to inclusions modeling combustion in porous media and processes of transmission of electrical impulses in nerve axons.     

Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry–Émery and Otto–Villani. Further, we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.