Schedule for: 18w5094 - Topics in the Calculus of Variations: Recent Advances and New Trends

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the 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.

We present a short-time existence result for the surface diffusion equation with an elastic term and without additional curvature regularization. We shall also discuss the asymptotic stability of strictly stable stationary sets.

We study the behavior of surface energies defined over couples (E,u) where E is a set and u is a density on the boundary of E. Such energies have been considered in the context of materials science for modelling surface diffusion in a way that takes into consideration explicitly the effect of the free atoms moving on the surface (adatoms) in a regime where the elastic energy is negligible. We discuss regular critical points, existence and uniqueness of minimizers and we characterize the relaxation of the energy functional in a suitable topology. Finally, we will present approximations with phase field and discrete energy. This is a work in collaboration with Marco Caroccia and Laurent Dietrich

In this talk we discuss the equilibrium problem for a curved dislocation line in a three-dimensional domain. As the core radius tends to zero, we derive an asymptotic expression to characterize the induced elastic energy. We then obtain the force on the dislocation line as the variation of this expression and identify the highest order terms explicitly. As a main ingredient, we present an explicit asymptotic formula for the induced elastic strain which depends on the curvature of the dislocation line and thus highlights the difference with existing work on straight dislocation lines.

We consider a shape optimization problem related to the design of polymer scaffolds for bone tissue engineering. Globally, bone loss due to trauma, osteoporosis, or osteosarcoma comprises a major reason for disability. To this day, autograft, i.e., a graft of bone tissue from a different place in the same body, remains the gold standard for large scale bone loss. This is despite major issues, for example donor site morbidity and limited availability. An ideal scaffold to be implanted in place of lost bone tissue must satisfy a number of different criteria, apart from the requirement of biocompatibility. It should be bioresorbable, so that no foreign objects remain after the regeneration time. In particular, however, during the regeneration time, it should provide adequate mechanical stability, while not preventing osteogenesis. Inspired by this biomechanical challenge, we consider the following shape optimization problem. In a periodic setting, by means of homogenization, one can obtain the effective elastic modulus for a given structure of a linearly elastic material occupying a set $E\subset \Omega=[0,1]^3$ under a certain loading condition. The objective is to find a set $E_\text{opt}$ (the occupied scaffold volume in a unit cell) such that the minimum of the effective elastic modulus for $E_\text{opt}$ and of the effective elastic modulus of the complement of the set $E^c_\text{opt}$ (i.e., of the regenerated bone tissue) is maximized. Joint work with Martin Rumpf and Stefan Simon (both Bonn).

A variational model for epitaxially-strained thin films deposited on substrates is derived by $\Gamma$-convergence from the so-called transition-layer model available in the literature. The regularity of energy-minimal film profiles is studied by establishing the internal-ball condition and by implementing some arguments from transmission problems. The possibility of different elastic properties between the film and the substrate is included in the analysis, as well as the surface tensions of all three involved interfaces: film/gas, substrate/gas, and film/substrate. The results relate to both the Stranski-Krastanow and the Volmer-Weber modes. Moreover, geometrical conditions are provided for the optimal wetting angle, i.e., the angle formed at the contact points between films and substrates. In particular, the Young-Dupr\`e law is shown to hold, yielding what appears to be the first analytical validation of such law in the context of Continuum Mechanics for a thin-film model. This is a joint work with Elisa Davoli (Vienna).

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!

Flowing a fluid is a familiar and efficient way to cool: fans cool electronics, water cools nuclear reactors, and the atmosphere cools the surface of the Earth. In this talk, we discuss a class of variational problems originating from fluid dynamics concerning the design of a wall-bounded incompressible flow to achieve optimal transport of heat between two rigid walls. Guided by a perhaps unexpected connection between this general class of optimal design problems and other more familiar "energy-driven pattern formation" problems from materials science, we construct nearly optimal flows featuring self-similar "branching" patterns in the advection-dominated limit. These patterns remind of (carefully designed versions) of the complex multi-scale patterns occurring in naturally turbulent fluids, but whether real atmospheric turbulence achieves optimal heat transport insofar as scaling laws are concerned remains a question of great theoretical interest. We address this question, proving that in Rayleigh's original two-dimensional model of Rayleigh-Benard convection between stress-free walls, optimal transport of heat is in fact not achieved. This is joint work with Charlie Doering (UM).

Non-Euclidean, or incompatible elasticity is an elastic theory for bodies that do not have a reference (stress-free) configuration. It applies to many systems, in which the elastic body undergoes inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold into another one of the same dimension, by minimizing an appropriate energy functional. This model is a source to many interesting analytic and geometric questions, and in this talk I will present some of them. Specifically, I will discuss the relation between the infimum elastic energy ("energetic incompatibility") and the curvature discrepancy between the manifolds ("geometric incompatibility"), and show how this question relates to a generalization of Reshetnyak's rigidity theorem to Riemannian manifolds. If time permits I will discuss more quantitative versions of this question. Based on joint work with Raz Kupferman and Asaf Shachar.

TBA

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

The Minimizing Movement (MM) scheme is a variational method introduced by E. De Giorgi to solve gradient flows in a quite general setting. In finite dimensional Euclidean spaces, when the driving function f is continuously differentiable, it is not difficult to see that all the limit curves are solutions to the ODE system generated by the gradient of f. However, since this vector field is only continuous, solutions may be not unique and there are solutions which cannot be obtained as a direct limit of the MM scheme. In his inspiring 1993 paper “New problems on Minimizing Movements”, De Giorgi raised the conjecture that all the solutions can be obtained as limit of a modifed scheme, obtained by a Lipschitz perturbation of f, converging to f in the Lipschitz norm as the time step goes to 0. We present some ideas of the proof of this conjecture related to nonsmooth calculus and to a new distinguished class of solutions to gradient flows. We will also discuss a partial extension of this result to infinite dimensional Hilbert spaces. In collaboration with Florentine Fleissner.

In this talk we will establish a quantitative rigidity estimate for two-well nonlinear elastic energies, in the case in which the two wells have exactly one rank-one connection, and we will analyze linearization around a laminate in the setting of two-dimensional solid-solid phase transition. In particular, we will identify by Gamma-convergence an effective limiting model. This limiting description will be given by the sum of a quadratic linearized elastic energy with two surface terms, corresponding to the total length of interfaces created by jumps of the deformation gradient and of the displacement field, respectively. This is joint work with Manuel Friedrich.

Using the finite volume method, one can define a discrete Kantorovich distance with a Riemannian structure based on a Euclidean mesh. We show that in most cases, the limit distance as mesh size tends to zero, in the sense of Gamma- or Gromov-Hausdorff-convergence, is strictly less than the standard Kantorovich distance. This is due to an oscillation effect reminiscent of homogenization. We introduce a geometric condition on the mesh that prevents oscillations and are able to show Gromov-Hausdorff convergence under this condition.

Liftings and their associated Young measures are new tools to study the asymptotic behaviour of sequences of BV-maps under weak* convergence. Their main feature is that they allow to keep track of the precise shape of the jump path and as such are natural objects whenever different ways of approaching a jump need to be distinguished. While this tool has several promising applications, in this talk I will focus on its use to prove lower semicontinuity for linear-growth functionals that depend on the value of the argument function, u(x), besides its gradient. It is well known that in this situation the particular shape of jumps cannot be neglected. Using the theory of liftings, we can prove relaxation theorems under essentially optimal assumptions, generalizing a classical theorem by Fonseca & Müller (1993). The key idea is that liftings provide the right way of localizing the functional in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow up procedure. This is joint work with Giles Shaw.

The theory of structured deformations shows good potential to deal with mechanical problems where multiple scales and fractures are present. Mathematically, it amounts to relaxing a given energy functional and to show also the relaxed one has an integral representation. In this seminar, I will focus on a problem for thin objects: the derivation of a 2D relaxed energy via dimension reduction from a 3D energy, incorporating structured deformations in the relaxation procedure. I will discuss the two-step relaxation (first dimension reduction, then structured deformations and viceversa) and I will compare it with another result in which the two relaxation procedures are carried out simultaneously. An explicit example for purely interfacial initial energies will complete the presentation. These results have been obtained in collaboration with G. Carita, J. Matias, and D.R. Owen.

We present a variant of the Cahn-Hilliard energy model for immiscible fluids which incorporates the effect of periodic heterogeneity at small scales. We describe the asymptotic behavior of minimizers using Gamma convergence methods. This talk will address some of the major difficulties in passing to a homogenized limit, which will require heavy exploitation of the self-symmetries of the N-dimensional integer lattice. This is a joint work with Riccardo Cristoferi and Irene Fonseca.

We study the mathematical well-posedness of the Crystalline Mean Curvature Flow in all dimensions and for arbitrary anisotropies and (convex) mobilities by means of a variational approach recently developed in collaboration with A. Chambolle, M. Novaga, and M. Ponsiglione.

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

We consider thin ferromagnetic films with strong perpendicular anisotropy. Experimental observation show the formation of bubble and stripe domain patterns in such ferromagnetic films. We investigate ground states and low energy states for the corresponding micromagnetic free energy functional. Using rigorous analysis, we identify to leading the critical scaling where the phase transition from single domain state to multi-domain states occurs. In the single domain regime, we derive a two dimensional effective model in the framework of Gamma convergence. For the multidomain regime, we derive a scaling law for the minimal energy and derive certain qualitative properties for minimizers of the system. This is joint work with C. Muratov and F. Nolte.

We prove that special functions of bounded deformation with small jump sets are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness of the jump set for minimizers of the Griffith energy.

Thin obstacle problems arise in the modelling of many chemical, physical and financial problems. In this talk I will present a robust strategy of proving optimal regularity for low regularity coefficients. Crucial technical ingrediets are a linearization technique based on Liouville theorems and the epiperimetric inequality. With the optimal regularity at hand, I will discuss quantitative higher regularity results for the thin obstacle problem in the presence of Hölder coefficients. Crucial ingredients in this context are a Hodograph-Legendre transform, the analysis of a fully nonlinear degenerate elliptic equation and the introduction of suitable function spaces. This is based on joint work with H. Koch and W. Shi.

This talk is devoted to confront two different approaches to the problem of dynamic perfect plasticity. Interpreting this model as a constrained boundary value Friedrichs' system enables one to derive admissible hyperbolic boundary conditions. Using variational methods, we show the well-posedness of this problem in a suitable measure theoretic setting. We prove that this unique variational solution actually coincides with the unique entropic solution of the hyperbolic formulation. Finally, thanks to the finite speed propagation property, we establish a new short time regularity result for the solution. This is a joint work with Clément Mifsud.

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

I will report on recent results with M. Jesenko (Univ. Freiburg) on a homogenization result for Hencky plasticity functionals with non-convex potentials. Also the influence of a small hardening parameter is investigated and it will be shown that homogenization and taking the vanishing hardening limit commute.

Epitaxy is a process in which a thin film is grown above a much thicker substrate. Even in the simplest case, with no deposition, and purely elastic interactions, such growth leads to a nonuniform film thickness since the film and the substrate can have different rigidity constants. The resulting system is thus an energy driven one, but quite irregular. Similarly, the evolution of nematic liquid crystals, systems is modeled by a highly complex energy driven system. In this talk I will present some recent results about the regularity of solutions to several equations arising from nematic liquid crystals and epitaxy.

We consider minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $C^{1, 1/2}$ on each side of the fracture. Moreover, we prove that near non-degenerate points the fracture set is $C^{1, \alpha}$, for some $\alpha \in (0,1)$. This is joint work with Luis Caffarelli and Alessio Figalli.

We explore a specific system in the mechanics of thin elastic​ sheets in which geometry and loading conspire to generate fine-scale wrinkling. This system -- a twisted ribbon held with small tension -- was examined experimentally by Chopin and Kudrolli [Phys Rev Lett 111, 174302, 2013]. There is a regime where the ribbon wrinkles near its center. A recent paper by Chopin, Démery, and Davidovitch models this regime using a von Kármán-like variational framework [J Elasticity 119, 137-189, 2015]. Our contribution is to give upper and lower bounds for the minimum energy as the thickness tends to zero. Since the bounds differ by a thickness-independent prefactor, we have determined how the minimum energy scales with thickness. Along the way we find estimates on Sobolev norms of the minimizers, which provide some information on the character of the wrinkling.

The Cahn-Hilliard model is a classical model for microscopic phase transitions in materials, which serves as an important prototype of many phase transition problems in continuum mechanics. I will discuss some recent results regarding second-order Gamma-convergence of these models using a novel rearrangement technique, as well as the application of these results to the study of metastable states for the Allen-Cahn equation. This is joint work with Giovanni Leoni and Matteo Rinaldi.

Fracture problems and damage processes have been widely studied in the last twenty years, starting from the pioneristic work of Griffith (and the later approach of Francfort - Marigò) to the more recent $Gamma$-convergence point of view developed by exploiting the Ambrosio - Tortorelli approximation for image recovery applied to the context of function of Bounded Deformation. In this talk we ill give a brief overview of the state of art of such a topic and we will end by presenting a recent approximation of fluid driven fracture based on a suitable correction of the classical energy with a lower order term that enable the introduction of several application. Some numerical experiments will also be discussed.

I will present an analysis of the sets that minimize the Gaussian perimeter plus the norm of the barycenter. These two terms are in competition, and in general the solutions are not the half-spaces. In fact we prove that when the volume is close to one, the solutions are the strips centered in the origin. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one. Co-Author: Vesa Julin

Phase field models are often used in situations where sharp interface simulations are difficult, in particular where topological changes can occur. While these are a feature in many applications, we may want to control at least the connectedness of either the phase transition regions or the phases. We present a method which allows us to approximate a relaxation of the perimeter functional under a connectedness constraint in two dimensions and Willmore's curvature energy at connected surfaces in three dimensions. We will give numerical evidence of the effectiveness of the method and describe how it can be implemented in an efficient manner.

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

I will present an overview of recent analytical developments in the studies of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. Due to the fundamental nature of Coulombic interaction, these problems arise in systems of very different physical nature and on vastly different scales: from femtometer scale of a single atomic nucleus to micrometer scale of droplets in electrosprays to kilometer scale of neutron stars. In the talk, I will focus on Rayleigh's model of perfectly conducting liquid drops and discuss the basic questions of existence vs. non-existence, as well as some qualitative properties of global energy minimizers in these models.

Graphene is locally two-dimensional but not flat. Nanoscale ripples appear in suspended samples and rolling-up often occurs when boundaries are not fixed. In this talk, I explain this variety of graphene geometries by classifying all ground-state deformations of the hexagonal lattice with respect to configurational energies including two- and three-body terms. We prove that all energy minimizers are either periodic in one direction, as in the case of ripples, or rolled up, as in the case of nanotubes. For suspended samples we refine the analysis further and prove the emergence of wave patterning. Specifically, we show that almost minimizers of the configurational energy develop waves with certain wavelength, independently of the size of the sample. The talk is based on joint work with Ulisse Stefanelli.

In this talk we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the Alt-Caffarelli energy functional.

We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals defined on space-time.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.