Schedule for: 23w5018 - Compensated Compactness and Applications to Materials

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the critical case case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed, now leading to an isotropic interfacial energy. In the subcritical case with moving wells, where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is observed that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains. This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), USA), based on previous results also obtained with Adrian Hagerty (USA) and Cristina Popovici (USA).

This talk will review the fully nonlinear system (geometric and material) of Field Dislocation Mechanics to establish an exact analogy with the equations of ideal magnetohydrodynamics (ideal MHD) under suitable physically simplifying circumstances. Weak solutions with various conservation properties have been established for ideal MHD recently by Faraco, Lindberg, and Szekelyhidi (2021) using the techniques of compensated compactness and convex integration; by the established analogy, these results would seem to transfer directly to the idealization of Field Dislocation Mechanics considered. A dual variational principle will be demonstrated for this system of PDE.

This talk adresses the question of the interplay between relaxation and irreversibility through evolution processes in damage mechanics, by inquiring the following question: can the quasi-static evolution of an elastic material undergoing a process of plastic deformation be derived as the limit model of a sequence of quasi-static brittle damage evolutions ? This question is motivated by the static analysis by Babadjian, Iurlano and Rindler who have shown how a brittle damage model introduced by Francfort and Marigo can lead to a model of (Hencky) perfect plasticity. Problems of damage mechanics being rather described through evolution processes, it is natural to extend this analysis to quasi-static evolutions, where the inertia is neglected. We consider the case where the medium is subjected to time-dependent boundary conditions, in the one-dimensional setting. The idea is to combine the scaling law introduced by Babadjian, Iurlano and Rindler with the quasi-static brittle damage evolution introduced by Francfort and Garroni, and try to understand how the irreversibility of the damage process will be expressed in the limit evolution. Surprisingly, the interplay between relaxation and irreversibility of the damage is not stable through time evolutions. Indeed, depending on the choice of the prescribed Dirichlet boundary condition, the effective quasi-static damage evolution obtained may not be of perfect plasticity type.

Whenever the stored energy density of a hyperelastic material has slow growth at infinity (below |F|^p with p less than the space dimension), it may undergo cavitation (the nucleation and sudden growth of internal voids) under large hydrostatic tension [Ball, 1982; James & Spector, 1992]. This constitutes a failure of quasiconvexity and, hence, a challenge for the existence theory in elastostatics [Ball & Murat, 1984]. The obstacle has been overcome under certain coercivity hypotheses [Müller & Spector, 1995; Sivaloganathan & Spector, 2000] which, however, fail to be satisfied by the paradigmatic example in elasticity: that of 3D neo-Hookean materials. A joint work with Marco Barchiesi, Carlos Mora-Corral, and Rémy Rodiac will be presented, where this borderline case was solved for hollow axisymmetric domains. Partial results leading to a solution when the axis of rotation is contained (where the dipoles found by [Conti & De Lellis, 2003] must be proved to be non energy-minimizing) will also be discussed.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We will outline some elementary questions and theorems about differential inclusions. Then make a "discontinuous jump" and talk about the concept of entropies from scalar conservation laws and the adaption of this concept to the Aviles Giga functional. Then we show how these topics connect and how the connection has application to both differential inclusions and to the Aviles Giga.

In this talk we discuss the homogenisation of vectorial integral functionals with $q$ growth in a bounded domain of $\mathbb R^n$, $n>q>1$, which is perforated by a random number of small spherical holes with random radii and centres. We show that for a class of stationary short-range correlated processes for the centres and radii of the holes, in the homogenised limit we obtain a nonlinear averaged analogue of the ``strange term'' obtained by Cioranescu and Murat in 1982, in the periodic case. In our case we only require that the random radii have finite $n-q$-moment, which is the minimal assumption to ensure that the expectation of the nonlinear capacity of the balls is finite. Although under this assumption there are holes which overlap with probability one, we can prove that the clustering holes do not have any impact on the homogenisation procedure and the limit functional. This is a joint work with K. Zemas (University of Muenster) and L. Scardia (Heriot Watt University).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We characterize the passage from nonlinear to linearized Griffith-fracture theories under non-interpenetration constraints. In particular, sequences of deformations satisfying a Ciarlet-Necas condition in $SBV^2$ and for which a convergence of the energies is ensured, are shown to admit asymptotic representations in $GSBD^2$ satisfying a suitable contact condition. With an explicit counterexample, we prove that this result fails if convergence of the energies does not hold. We further prove that each limiting displacement satisfying the contact condition can be approximated by an energy-convergent sequence of deformations fulfilling a Ciarlet-Necas condition. The proof relies on a piecewise Korn-Poincaré inequality in $GSBD^2$, on a careful blow-up analysis around jump points, as well as on a refined $GSBD^2$-density result guaranteeing enhanced contact conditions for the approximants. This is joint work with Stefano Almi (Naples) and Manuel Friedrich (Erlangen).

In this talk I will present some results obtained with B. Merlet in recent years on a family of energies penalizing oscillations in oblique directions. These functionals, which first appeared in the study of an isoperimetric problem with non-local interactions, can be seen as a natural extension of the Bourgain-Brezis-Mironescu energies. A central insight is that these energies actually control second order derivatives rather than first order ones. Indeed, functions of finite energy have mixed (or oblique) derivatives given by bounded measures. The main focus of the talk is the study of the rectifiability properties of these 'defect' measures. Time permitting we will draw connections with branched transportation, PDE constrained measures and Aviles-Giga type differential inclusions.

One of the seminal contributions of Murat and Tartar to the theory of compensated compactness was to observe how a linear PDE constraint may prevent the formation of wild oscillations in the directions where the PDE is highly regularizing on laminate structures. For example, it is well-known that gradients can only oscillate in laminates of rank-one connected matrices. In fact, we know now that this is the reason why Alberti’s rank-one theorem holds for gradient measures. This is perhaps one of the key insights revealed by the work of De Philippis and Rindler, who discovered that a linear PDE constraint imposes a strong pointwise constraint on the singular part of measures: the polar density belongs to the wave cone. In this talk, I will talk about how PDE constraints interact with the formation of mass concentrations (which are necessary for the formation of measure singularities). In particular, I will discuss how the formation of “strong mass concentrations” along a sequence of PDE-constrained functions is “fully unconstrained” as long as the expectation of its values belongs to the wave cone associated with the PDE. If time permits, I will explain the “gluing technique” behind the proof as well as some interesting applications.

Quasiconvexity is the fundamental existence condition for variational problems, yet it is poorly understood. Two outstanding problems remain: 1) does rank-one convexity, a simple necessary condition, imply quasiconvexity in two dimensions? 2) can one prove existence theorems for quasiconvex energies in the context of nonlinear Elasticity? In this talk we show that both problems have a positive answer in a special class of isotropic energies. Our proof combines complex analysis with the theory of gradient Young measures. On the way to the main result, we establish quasiconvexity inequalities for the Burkholder function which yield, in particular, many sharp higher integrability results. The talk is based on joint work with Kari Astala, Daniel Faraco, Aleksis Koski and Jan Kristensen.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the Willmore functional on graphs, with an additional penalization of the area where the curvature is non-zero. Interpreting the penalization parameter as a Lagrange multiplier, this corresponds to the Willmore functional with a constraint on the area where the graph is flat. Sending the penalization parameter to infinity and rescaling suitably, we derive the limit functional in the sense of Gamma-convergence.

G. Bouchitté formulated the "Vanishing Mass Conjecture" about twenty years ago, in the context of optimization of light elastic structures. Since then the only progress has been obtained by J.F. Babadjian, F. Iurlano and F. Rindler in 2021. In this talk I will illustrate this conjecture placing the emphasis on its geometric nature, and some partial results obtained with Andrea Marchese (University of Trento) and Andrea Merlo (University of the Basque Country).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

An important problem that often arises in shape optimization is the representation of the so-called Blaschke-Santal\'o diagram between two quantities related to a PDE on a varying domain. I will discuss in detail the case of torsional rigidity and first eigenvalue of the Laplace operator, even if in the literature several other cases have been considered. From the numerical point of view this amounts to represent the image of a given map $F:X\to\mathbb R^k$ (usually $k=2$), being $X$ a compact metric space. The case when $X$ is a subset of an Euclidean space $\mathbb R^d$ (with $d$ much larger than $k$) is also interesting and a suitable use of Voronoi tessellations plays an important role. This last research is in collaboration with Benjamin Bogosel and Edouard Oudet.

We present a nonoriented version of the Aviles-Giga functional which arises as a model for pattern formation (more precisely, for striped patterns in 2D). We show that sequences with uniformly bounded energy as the scale parameter goes to 0 are relatively compact in $L^1_{\rm loc}$. We also completely describe the limit configurations in the vanishing energy limit case. These result parallel their counterparts for the classical Aviles--Giga functional but new phenomena appear in the non oriented case and the proofs require new ideas. Joint work with Michael Goldman, Marc Pegon and Sylvia Serfaty.

Motivated by new nonlocal models in hyperelasticity, we discuss a class of variational problems with integral functionals depending on nonlocal gradients that correspond to truncated versions of the Riesz fractional gradient. We address several aspects regarding the existence theory of these problems and their asymptotic behavior. Our analysis relies on suitable translation operators that allow us to switch between the three types of gradients: classical, fractional, and nonlocal. These provide helpful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, the natural convexity notion in the classical calculus of variations, characterizes weak lower semicontinuity also in the fractional and nonlocal setting. As a consequence of a general Gamma-convergence statement, we derive relaxation and homogenization results. The analysis of the limiting behavior as the fractional order tends to 1 yields localization to a classical model. This is joint work with Javier Cueto (University of Nebraska-Lincoln) and Hidde Schönberger (KU Eichstätt-Ingolstadt).

Thin elastic sheets form various patterns when confined, from oscillatory wrinkling to concentrated folding (not to mention random creasing, as in crumpled paper). This talk will report recent progress towards an effective variational theory for wrinkles and folds. In the first part, we discuss the wrinkling of curved shallow shells that float on top of a flat water bath. By deriving and solving the Gamma limit of a rescaled shallow shell model, we explain the patterns that arise in a given floating shell. Here, a surprise is that the wrinkles of positively and negatively curved shells turn out to be linked. We then discuss recent work on folds. Using a fully nonlinear model for a plate confined to a planar cavity and laterally squeezed, we prove a scaling law and compactness result involving a $BV$-type energy that arises to control the length of the folds. The limiting mid-plane is shown to deform by a length-preserving map that can change orientation across a singular set containing the folds. The first part is in collaboration with the physicists Eleni Katifori (U Penn) and Joey Paulsen (Syracuse), and the second part is with Samuel Wallace (UIC).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the classical theory, given a vector field $b$ on $\mathbb R^d$, one usually studies the transport/continuity equation driven by $b$ looking for solutions in the class of functions (with certain integrability) or at most in the class of measures. In this seminar I will talk about recent efforts, motivated by the modelling of defects in crystals, aimed at extending the previous theory to the case where the unknown are instead $k$-currents in $\mathbb R^d$, i.e. generalised $k$-dimensional surfaces. The resulting equation involves the Lie derivative $L_b$ of currents in direction $b$ and reads $\partial_t T_t + L_b T_t = 0$. I will prove existence and uniqueness for this equation in the class of normal currents, under the natural assumption of Lipschitz regularity of the vector field $b$. The argument relies crucially on the notion of decomposability bundle introduced recently by Alberti and Marchese and, in the particular case of $0$-currents, it yields a new proof of the uniqueness for the continuity equation in the class of signed measures. Based on joint works with G. Del Nin and F. Rindler.

In this talk we study the asymptotic behaviour of a general class of singularly-perturbed elliptic functionals of Ambrosio-Tortorelli type as the perturbation parameter vanishes. Under mild reg- ularity assumptions and suitable super-linear growth conditions on the integrands we show that the functionals $\Gamma$-converge (up to subsequences) to a free-discontinuity functional of brittle type. A key ingredient consists in providing asymptotic formulas for the limiting volume and surface integrands, which show that the volume and surface contributions decouple in the limit. The abstract $\Gamma$-convergence result is then applied to the homogenisation of Ambrosio-Tortorelli type functionals. The talk is based on joint works with T. Esposito, R. Marziani, and C. I. Zeppieri.

Dislocation motion is a key feature of crystal plasticity at the smallest scales, and many mathematical challenges must be overcome to establish a well-posed theory which accurately couples dislocation motion and continuum theories in a three-dimensional nonlinear setting. This talk will present joint work with Filip Rindler on a model which we believe has the physical richness and the right mathematical structure necessary for a model which is both physically meaningful and well-posed close to the microscale. In particular, we propose a novel geometric language built on the concepts of space-time currents, or "slip trajectories" and the "crystal scaffold" to describe the movement of discrete dislocations. In order to place our model into context, we further show that it recovers several laws that were known in special cases before, such as the equation for the Peach-Koehler force on a dislocation in a linearised context.

The Bernstein problem asks whether entire minimal graphs in $\mathbb R^{n+1}$ are necessarily hyperplanes. It is known through spectacular work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti that the answer is positive if and only if $n < 8$. The anisotropic Bernstein problem asks the same question about minimizers of parametric elliptic functionals, which are natural generalizations of the area functional that both arise in material science and offer important technical challenges. We will discuss the recent solution of this problem (the answer is positive if and only if $n < 4$). This is joint work with Y. Yang.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Composite materials display a wide range of conduction properties depending on the geometric configuration of the phases. A classical, but mostly unsolved problem, is to find the range of the effective conductivity, a constant but in general anisotropic matrix describing the overall electrical behavior of the composite (sometimes called G-closure). The strategy is to establish bounds that do not depend on the microgeometry and then to exhibit optimal configurations, i.e., arrangements that saturate a given bound. Optimal bounds for isotropic mixtures are known in a few cases, beginning with the pioneering work of [Hashin and Shtrikman 1962]. However, the complete G-closure is known only in the special case of two isotropic phases [Murat & Tartar 1985, Lurie & Cherkaev 1986]. Building on previous works by [Milton & Nesi 1990], we provide new anisotropic optimal microgeometries in the case of a three dimensional polycrystalline material, where the principal conductivities of the basic crystal are given, but the orientation of the crystal is allowed to change from point to point. (Joint work with Nathan Albin and Vincenzo Nesi.)

In a model of many particles interacting in $\mathbb R^d$, like in Density Functional Theory or crowd motion, the energy cost is usually taken to be repulsive and described by a two-point function $c_\varepsilon(x,y) =\ell(\frac{|x-y|}{\varepsilon})$ where $\ell: \mathbb R_+ \to [0,\infty]$ is decreasing to zero at infinity and small parameter $\varepsilon>0$ scales the interaction distance. After reviewing some link with multimarginal optimal transport, I will focus on new results obtained in collaboration with R.Mahadevan (university of Concepcion, Chili) where we identify the mean-field energy of such a model in the short-range regime $\varepsilon\ll 1$ assuming merely that $\int_{r_0}^\infty \ell(r) r^{d-1}\, dr <+\infty$. This extends (and simplifies) existing results in the homogeneous case $\ell(r) = r^{-s}$ where $s>d$.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this joint work with JF Babadjian we investigate a typical functional of the gradient which exhibits linear growth at infinity. The relaxed functional has BV minimizers. In 2d, their uniqueness is intimately tied to the properties of a spatial continuity equation for which the theory of regular Lagrangian flows does unfortunately not apply. However, techniques related to the work of Jabin-Otto-Perthame on 2d Ginzburg-Landau models permit a better understanding of the associated characteristic flow from which uniqueness hopefully follows.

I will present a derivation of a line tension model for dislocations in 3D starting from a variational model which accounts for the elastic energy induced by incompatible elastic fields. Under a kinematic constraint that forces the dislocations of being diluted on a mesoscopic scale, via Gamma-convergence, we deduce energies concentrated on 1-rectifiable lines which can be interpreted as line tension energies for dislocations in a sigle crystal. The result is based on a recent paper in collaboration with S. Conti and R. Marziani. We treat a quite general framework which includes several different regularised variational models present in the literature, ranging from linear elastic energies with core regularization to non linear elastic energies with sub-quadratic regularisation.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.