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Raz Kupferman

Publications and source records attributed to Raz Kupferman.

At least 19 recordsLinked to original sources
arXiv2026

Energy scaling laws for thin elastic sheets with topological defects

We derive energy scaling laws for thin elastic sheets with topological defects --- disclinations and dislocations --- for a fully nonlinear 3D model. For disclinations, the scaling laws are tight in the thickness parameter, and improve upon previous results by applying simultaneously to positive and negative disclinations (e-cones) and by giving an explicit dependence on the defect parameter; this latter dependence is, however, still not tight. For thin bodies with dislocations, these are, to the best of our knowledge, the first rigorous bounds for models of finite thickness, are tight when the Burgers vector is not large with respect to the thickness, and relate to a well-known conjecture from the physics literature about the scaling. A main tool is modeling these bodies in the framework of non-Euclidean elasticity, as bodies with a curl-free pre-strain; the curl-freeness allows us to obtain geometric rigidity estimates for the lower bounds.

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arXiv2026

Branch points and non-density for finite-bending isometric immersions of hyperbolic surfaces

The space of $W^{2,2}$-isometric immersions of a surface into $\mathbb{R}^3$ arises naturally in the variational theory of thin elastic sheets: it is precisely the finite-bending class, where the bending energy --- the $L^2$-norm of the second fundamental form --- is finite. For sheets with negative Gaussian curvature, previous work has identified branch points, where "too many" asymptotic directions meet --- or, equivalently, where the index of the Gauss map is not $-1$ --- as a potentially important mechanism in shape selection and pattern formation. Such branch points are precluded for $C^2$-isometric immersion. We show that this index-based notion of branch points extends to the full finite-bending class: Namely, for every $W^{2,2}$ isometric immersion of a negatively-curved surface, the index of the Gauss map is well-defined at every point, and the set of branch points is discrete. We further show that the index is stable under $W^{2,2}$-convergence, and thus, an isometric immersion with branch points cannot be approximated by $C^2$-isometric immersions. Conversely, we show that every negatively-curved metric locally admits $W^{2,2}$-isometric immersions (in fact, $C^{1,1}$) with branch points of arbitrary order. Consequently, $C^2$-isometric immersions are, in general, not dense among finite-bending ones, in stark contrast with the flat and positively curved cases.

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arXiv2025

The Willmore energy and curvature concentration

We study isometric immersions of a Riemannian surface $(\Omega,\frak{g})$, where $\Omega \subset \mathbb{R}^2$, into $\mathbb{R}^3$. We consider their bending energy, i.e., the square of the $L^2$-norm of their second fundamental form, which is equivalent to the Willmore functional. We obtain two new lower bounds for this energy, one in terms of the Gaussian curvature of the surface, and the other in terms of a Burgers vector -- a measure of non-flatness connected to torsion. These new estimates provide optimal blowup rates of the energy when the curvature is concentrated (e.g., in a conical geometry). In the more subtle case of dipoles of concentrated curvature, we use the Burgers vector estimates to obtain an optimal blowup rate in terms of the size of the system. Our motivation comes from non-Euclidean elasticity, in which cones and curvature-dipoles play a central role. The lower bounds derived in this work directly yield lower bounds for the elastic energy of thin elastic sheets. The derivation of the curvature-based lower bound involves an isoperimetric inequality for framed loops, which we believe to be of independent interest.

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arXiv2024

Linearization in incompatible elasticity for general ambient spaces

Motivated by recent interest in elastic problems in which the target space is non-Euclidean, we study a limit where local rest distances within an elastic body are incompatible, yet close to, distances within the ambient space. Specifically, we obtain, via $\Gamma$-convergence, a limit elastic model for a sequence of elastic bodies $(M,g_\varepsilon)$ in an ambient space $(S,s)$, for Riemannian metrics $g_\varepsilon$ and $s$ such that $g_\varepsilon \to s$. Furthermore, we relate the minimum of the limit problem to a linearized curvature discrepancy between $g_\varepsilon$ and $s$, using recent results of Kupferman and Leder. This relation confirms a linearized version of a long-standing conjecture in elasticity regarding the relation between the elastic energy and the curvature of the underlying space. The main technical challenge, compared to other linearization results in elasticity, is obtaining the correct notion of displacement for manifold-valued configurations, using Sobolev truncations and parallel transport. We show that the associated compactness result is obtained if $(S,s)$ satisfies a quantitative rigidity property, analogous to the Friesecke--James--M\"uller rigidity estimate in Euclidean space, and show that this property holds when $(S,s)$ is a round sphere.

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arXiv2023

Stability of isometric immersions of hypersurfaces

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$-perturbations of their fundamental forms: For a manifold $M^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:M^d\to N^{d+1}$, whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result by the authors to a general target manifold $N$, removing a constant curvature assumption. The method of proof differs from that in Alpern et al.: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to Ciarlet et al. (Anal. Appl. 2019) but with no a-priori assumed bounds.

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arXiv2023

Elliptic Pre-Complexes, Hodge-like Decompositions and Overdetermined Boundary-Value Problems

We solve a problem posed by Calabi more than 60 years ago, known as the Saint-Venant compatibility problem: Given a compact Riemannian manifold, generally with boundary, find a compatibility operator for Lie derivatives of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and to their inherent gauge freedom. To this end, we develop a framework generalizing the theory of elliptic complexes for sequences of linear differential operators $(A_{\bullet})$ between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions, and the order of $A_{k+1}A_k$ does not exceed the order of $A_k$. We show that every elliptic pre-complex $(A_{\bullet})$ can be "corrected" into a complex $(\mathcal{A}_{\bullet})$ of pseudodifferential operators, where $\mathcal{A}_k - A_k$ is a zero-order correction within this class. The induced complex $(\mathcal{A}_{\bullet})$ yields Hodge-like decompositions, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. We apply the theory on elliptic pre-complexes of exterior covariant derivatives of vector-valued forms and double forms satisfying generalized algebraic Bianchi identities, thus resolving a set of compatibility and gauge problems, among which one is the Saint-Venant problem.

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arXiv2023

From Volterra dislocations to strain-gradient plasticity

We rigorously derive a strain-gradient model of plasticity as a $\Gamma$-limit of continuum bodies containing finitely-many edge-dislocations (in two dimensions). The key difference from previous such derivations is the elemental notion of a dislocation: we work in a continuum framework in which the lattice structure is represented by a smooth frame field, and the presence of a dislocation manifests in a circulation condition on that frame field; the resulting model is a Lagrangian approach with a multiplicative strain decomposition. The multiplicative nature of the geometric incompatibility generates many technical challenges, which require a systematic study of the geometry of bodies containing multiple dislocations, the definition of new notions of convergence, and the derivation of new geometric rigidity estimates pertinent to dislocated bodies. Our approach places the strain-gradient limit in a unified framework with other models of dislocations, which cannot be addressed within the "admissible strain" approach used in previous works.

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arXiv2021

On Saint-Venant compatibility and stress potentials in manifolds with boundary and constant sectional curvature

We address three related problems in the theory of elasticity, formulated in the framework of double forms: the Saint-Venant compatibility condition, the existence and uniqueness of solutions for equations arising in incompatible elasticity, and the existence of stress potentials. The scope of this work is for manifolds with boundary of arbitrary dimension, having constant sectional curvature. The central analytical machinery is the regular ellipticity of a boundary-value problem for a bilaplacian operator, and its consequences, which were developed in [KL21]. One of the novelties of this work is that stress potentials can be used in non-Euclidean geometries, and that the gauge freedom can be exploited to obtain a generalization for the biharmonic equation for the stress potential in dimensions greater than two.

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arXiv2021

Double forms: Regular elliptic bilaplacian operators

Double forms are sections of the vector bundles $\Lambda^{k}T^*\mathcal{M}\otimes \Lambda^{m}T^*\mathcal{M}$, where in this work $(\mathcal{M},\mathfrak{g})$ is a compact Riemannian manifold with boundary. We study graded second-order differential operators on double forms, which are used in physical applications. A Combination of these operators yields a fourth-order operator, which we call a double bilaplacian. We establish the regular ellipticity of the double bilaplacian for several sets of boundary conditions. Under additional conditions, we obtain a Hodge-like decomposition for double forms, whose components are images of the second-order operators, along with a biharmonic element. This analysis lays foundations for resolving several topics in incompatible elasticity, most prominently the existence of stress potentials and Saint-Venant compatibility.

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arXiv2020

Asymptotic rigidity for shells in non-Euclidean elasticity

We consider a prototypical "stretching plus bending" functional of an elastic shell. The shell is modeled as a d-dimensional Riemannian manifold endowed, in addition to the metric, with a reference second fundamental form. The shell is immersed into a (d+1)-dimensional ambient space, and the elastic energy accounts for deviations of the induced metric and second fundamental forms from their reference values. Under the assumption that the ambient space is of constant sectional curvature, we prove that any sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell into ambient space, having the reference second fundamental form. In particular, if the ambient space is Euclidean space, then the reference metric and second fundamental form satisfy the Gauss-Codazzi-Mainardi compatibility conditions. This theorem can be viewed as a (manifold-valued) co-dimension 1 analog of Reshetnyak's asymptotic rigidity theorem. It also relates to recent results on the continuity of surfaces with respect to their fundamental forms.

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arXiv2020

Limits of distributed dislocations in geometric and constitutive paradigms

The 1950's foundational literature on rational mechanics exhibits two somewhat distinct paradigms to the representation of continuous distributions of defects in solids. In one paradigm, the fundamental objects are geometric structures on the body manifold, e.g., an affine connection and a Riemannian metric, which represent its internal microstructure. In the other paradigm, the fundamental object is the constitutive relation; if the constitutive relations satisfy a property of material uniformity, then it induces certain geometric structures on the manifold. In this paper, we first review these paradigms, and show that they are equivalent if the constitutive model has a discrete symmetry group (otherwise, they are still consistent, however the geometric paradigm contains more information). We then consider bodies with continuously-distributed edge dislocations, and show, in both paradigms, how they can be obtained as homogenization limits of bodies with finitely-many dislocations as the number of dislocations tends to infinity. Homogenization in the geometric paradigm amounts to a convergence of manifolds; in the constitutive paradigm it amounts to a $Γ$-convergence of energy functionals. We show that these two homogenization theories are consistent, and even identical in the case of constitutive relations having discrete symmetries.

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arXiv2019

Variational Convergence of Discrete Geometrically-Incompatible Elastic Models

We derive a continuum model for incompatible elasticity as a variational limit of a family of discrete nearest-neighbor elastic models. The discrete models are based on discretizations of a smooth Riemannian manifold $(M,\mathfrak{g})$, endowed with a flat, symmetric connection $\nabla$. The metric $\mathfrak{g}$ determines local equilibrium distances between neighboring points; the connection $\nabla$ induces a lattice structure shared by all the discrete models. The limit model satisfies a fundamental rigidity property: there are no stress-free configurations, unless $\mathfrak{g}$ is flat, i.e., has zero Riemann curvature. Our analysis focuses on two-dimensional systems, however, all our results readily generalize to higher dimensions.

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arXiv2018

Homogenization of edge-dislocations as a weak limit of de-Rham currents

In the material science literature we find two continuum models for crystalline defects: (i) A body with (finite) isolated defects is typically modeled as a Riemannian manifold with singularities, and (ii) a body with continuously distributed defects, which is modeled as a smooth (non-singular) Riemannian manifold with an additional structure of an affine connection. In this work we show how continuously distributed defects may be obtained as a limit of singular ones . The defect structure is represented by layering 1-forms and their singular counterparts - de-Rham (n-1) currents. We then show that every smooth layering $1$-form may be obtained as a limit, in the sense of currents, of singular layering forms, corresponding to arrays of edge dislocations. As a corollary, we investigated manifolds with full material structure, i.e., a complete co-frame for the co-tangent bundle. We define the notion of singular torsion current for manifolds with a parallel structure and prove its convergence to the regular smooth torsion tensor at homogenization limit. Thus establishing the so-called emergence of torsion at the homogenization limit.

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arXiv2018

Covariant Linearization of elasticity

In this paper we derive a general linearized theory for first-order continuum dynamics on manifolds with particular application to incompatible elasticity. We adopt a global approach viewing the equations of motion as a $1$-form on the configuration space which is the Banach manifold of $C^1$ time-dependent embeddings of a body manifold $\B$ into a space manifold $§$. The linearization is done by differentiating the equations 1-form with respect to an affine connection which we construct and study extensively. We provide detailed coordinate computations for the linearized equations of a large class of problems in continuum dynamics on manifolds.

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arXiv2018

Reshetnyak rigidity for Riemannian manifolds

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theory for conformal maps; we give a new, simple proof, by generalizing the Piola identity for the cofactor operator. Second, we prove that if there exists a sequence of mapping $f_n:M\to N$, whose differentials converge in $L^p$ to the set of orientation-preserving isometries, then there exists a subsequence converging to an isometric immersion. These results are generalizations of celebrated rigidity theorems by Liouville (1850) and Reshetnyak (1967) from Euclidean to Riemannian settings. Finally, we describe applications of these theorems to non-Euclidean elasticity and to convergence notions of manifolds.

math.DG↗
arXiv2018

A geometric perspective on the Piola identity in Riemannian settings

The Piola identity $\operatorname{div} \operatorname{cof} \nabla f=0$ is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: one follows the lines of the classical Euclidean derivation and the other is based on a variational interpretation via Null-Lagrangians. In both cases, we first review the Euclidean case before proceeding to the general Riemannian setting.

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arXiv2018

The emergence of torsion in the continuum limit of distributed edge-dislocations

We present a rigorous homogenization theorem for distributed dislocations. We construct a sequence of locally-flat Riemannian manifolds with dislocation-type singularities. We show that this sequence converges, as the dislocations become denser, to a flat non-singular Weitzenböck manifold, i.e. a flat manifold endowed with a metrically-consistent connection with zero curvature and non-zero torsion. In the process, we introduce a new notion of convergence of Weitzenböck manifolds, which is relevant to this class of homogenization problems.

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