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Carnegie Mellon University

Department of Mathematical Sciences


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Time Monday,
July 18
July 19
July 20
8:00-8:40 Registration/Breakfast Breakfast Breakfast
8:30-8:40 Welcome remarks
8:40-9:20 Ball Slemrod Friesecke
9:25-10:05 Ambrosio Dolbeault Marini
10:10-10:30 Coffee Break
10:30-11:10 Otto Luskin Chipot
11:15-11:55 Brezis Du Brezzi
12:00-12:10 LUNCH Poster Prizes Ceremony
12:10-1:30 LUNCH
1:30-2:10 DiBenedetto James Bauman
2:15-2:55 Carrillo Barmak Schwab
3:00-3:20 Coffee Break
3:20-4:00 Matano Xu Golovaty
4:05-4:45 Kohn Monsaingeon Bronsard
5:00-5:40   Poster Blitz  
6:00-8:00 Birthday Reception  


► Luigi Ambrosio
Title: Well Posedness of ODE's and Continuity Equations with Non Smooth Vector Fields, and Applications
Abstract: I will do a survey talk on this fascinating research topic, initiated by a seminal paper by DiPerna-Lions at the end of the '80. Since then the theory developed in many directions, that I will partly cover in my talk: minimal regularity assumptions on the vector field, quantitative error and convergence estimates, regularity of the flow. I will also cover more recent results: an application of the theory to the Vlasov-Poisson equation and the abstract side of the theory, which provides a canonical and stable notion of flow in metric measure structures.
► John Ball
Title: Jump Conditions and Polycrystals
Abstract: The talk will describe joint work with Carsten Carstensen on generalizations of the Hadamard jump condition for arbitrary Lipschitz mappings. Possible applications include nonclassical austenite-martensite interfaces and implications of compatibility across grain boundaries in polycrystals. For the latter a geometric description of polycrystals is required so as to exclude pathological geometry. An example of the use of a generalized jump condition to microstructure in a bi-crystal is given.
► Katayun Barmak
Title: Puzzle of Grain Growth Stagnation in Metallic Films: Simulations and Experiments
Abstract: Experimental grain growth characteristics are examined for Al and Cu films. The experimental data set is used to evidence stagnation of grain growth and arrive at a universal size distribution. This distribution differs from the distribution obtained in two-dimensional simulations of grain growth with isotropic boundary energy. Surface and elastic strain energy, anisotropy of boundary energy, grooving, impurity drag, and triple-junction drag are examined as the causes of the observed difference between experiment and simulation.
► Patricia Bauman
Title: Regularity and Eigenvalue Properties of Minimizers for Maier-Saupe Energies in Liquid Crystals
Abstract: We investigate regularity and properties of eigenvalues for minimizers of the Maier-Saupe energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Majumdar's modification of the Landau-de Gennes Q-tensor model model, so as to constrain the competing states to take values pointwise almost-everywere in the closure of a physically realistic range. We prove that minimizers are regular and in several model problems we use this regularity to prove that minimizers have values strictly within the physical range.
► Emmanuele DiBenedetto
Title: A Necessary and Sufficient Condition for the Continuity of Local Minima of Parabolic Variational Integrals with Linear Growth
Abstract: For proper minimizers of parabolic variational integrals with linear growth with respect to Du, we establish a necessary and sufficient condition for u to be continuous at a point, in terms of a sufficient fast decay of the total variation of u. These minimizers arise also as variational solutions to the parabolic 1-laplacian equation. Hence, the continuity condition continues to hold for such solutions.
► Haim Brezis
Title: Old-New Perspectives on the Winding Number. Where Cauchy, Fourier, Sobolev and Bourgain Meet
Abstract: This lecture deals with various recent developments concerning the topological degree of maps from the circle into itself. I will first explain how it can be extended beyond the class of continuous maps. This led to the "accidental" discovery of a simple, but intriguing formula connecting the degree of a map to its Fourier coefficients. The relation is easily justified when the map is smooth. However, the situation turns out to be extremely delicate if one assumes only continuity, or even Holder continuity. This "marriage" is more difficult than expected and there are many difficulties in this couple such as the following question I raised: "Can you hear the degree of a map from the circle into itself?" I will also present estimates for the degree leading to the question: "How much energy do you need to produce a map of given degree?". Many simple looking problems remain open. The initial motivation for this research came from the analysis of the Ginzburg-Landau model in Physics.
► Franco Brezzi
Title: Various VEM spaces and applications
Abstract: We discuss the definition of Virtual Elements for the approximation of different types of functional spaces (as $H({\rm div})$-conforming, $H({\rm curl})$-conforming, or even $H^k$-conforming spaces for $k\ge 2$) and their possible use in different applications. Note that, for a given decomposition of the computational domain into polytopal elements, the above spaces (as for classical $H^1$-conforming VEMs) are made of solutions of (systems of) PDEs in each element. Here too, the name of the game is to avoid solving these equations: not even in an approximate way. In particular we will concentrate on: the computation of the $L^2$-projection of the elements of VEM spaces on (vector valued) polynomials, on the use of this projection for the definition of suitable (and computable) scalar products, and on the use of these scalar products for solving PDE problems of interest in applications.
► Lia Bronsard
Title: Minimizers of the Landau-de Gennes Energy Around a Spherical Colloid Particle
Abstract: We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau-de~Gennes model. The nematic is assumed to occupy the exterior of a ball $B_{r_0}$, and satisfy homeotropic weak anchoring at the surface of the colloid and approach a uniform uniaxial state as $|x|\to\infty$. We study the minimizers in two different limiting regimes: for balls which are small $r_0\ll L^{\frac12}$ compared to the characteristic length scale $L^{\frac 12}$, and for large balls, $r_0\gg L^{\frac12}$. The relationship between the radius and the anchoring strength $W$ is also relevant. For small balls we obtain a limiting quadrupolar configuration, with a "Saturn ring" defect for relatively strong anchoring, corresponding to an exchange of eigenvalues of the $Q$-tensor. In the limit of very large balls we obtain an axisymmetric minimizer of the Oseen-Frank energy, and a dipole configuration with exactly one point defect is obtained.
This is joint work with Stan Alama and Xavier Lamy.
► José Antonio Carrillo de la Plata
Title: Diffusive Dominated Keller-Segel Model
Abstract: We analyse under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and attraction modelled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrisation techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as $t\to\infty$.
► Michel Chipot
Title: Asymptotic Issues in Infinite Cylinders
Abstract: We would like to present some results on the asymptotic behaviour of different problems set in cylindrical domains of the type $\ell \omega_1 \times \omega_2$ when $\ell \to \infty$. For $i=1, 2$ $\omega_i$ are two bounded open subsets in $\mathbb{R}^{d_i}$. To fix the ideas on a simple example consider for instance $\omega_1=\omega_2=(-1,1)$ and $u_\ell$ the solution to $$ -\Delta u_\ell = f ~~\text{in} ~~\Omega_\ell= (-\ell,\ell)\times (-1,1)~~,~~u_\ell = 0 ~~\text{on} ~~\partial \Omega_\ell. $$ It is more or less clear that, when $\ell \to \infty, $ $u_\ell $ will converge toward $u_\infty$ solution to $$ -\Delta u_\infty = f ~~\text{in} ~~\Omega_\infty= (-\infty,\infty)\times (-1,1)~~,~~u_\infty = 0 ~~\text{on} ~~\partial \Omega_\infty. $$ However this problem has infinitely many solutions since for every integer $k$ $$\text{exp}(k\pi x_1)\text{sin}(k\pi x_2)$$ is solution of the corresponding homogeneous problem. Our goal is to explain the selection process of the solution for different problems of this type when $\ell \to \infty$.
► Jean Dolbeault
Title: From Entropy Methods to Symmetry and Symmetry Breaking in Interpolation Inequalities
Abstract: Entropy methods have been introduced in the theory of nonlinear diffusion equations with the purpose of studying rates of convergence to equilibrium. Optimal rates of convergence have then been related with optimal constants in some interpolation inequalities, at least in some fundamental examples. The evolution equations themselves have been interpreted as gradient flows of the entropies with respect to an appropriate notion of distance, after the celebrated paper by R. Jordan, D. Kinderlehrer and F. Otto on the variational formulation of the Fokker-Planck equation. Hence it became clear that well chosen interpolation inequalities, which typically relate the entropies with their derivatives along the nonlinear flow, carry lots of useful informations for the solutions to the evolution problem. The reverse is also true and one can build adapted flows in order to study the cases of optimality in certain inequalities. The purpose of this lecture is to review some achievements in this direction and to illustrate the use of nonlinear diffusions as a tool with questions of uniqueness and symmetry breaking of the optimal functions in weighted inequalities that have been introduced by L. Caffarelli, R. Kohn and L. Nirenberg.
The latest result is a joint work with M.J. Esteban, M. Loss and M. Muratori.
► Qiang Du
Title: Localization of Nonlocal Continuum Models
Abstract: Recent development of nonlocal vector calculus and nonlocal calculus of variations provides a systematic mathematical framework for the analysis of nonlocal continuum models in the form of partial-integral equations. In this lecture, we discuss the localization of some nonlocal models and associated nonlocal function spaces in order to study connections with traditional local models given by partial differential equations and Sobolev spaces. In particular, we present some recent results on heterogeneous localization of nonlocal space including an extension of classical trace theorems to nonlocal spaces of functions with significantly weaker regularity. We also discuss their implications in nonlocal modeling of multiscale processes.
► Gero Friesecke
Title: Variational Problems and Multiscale Phenomena in Density Functional Theory
Abstract: Density functional theory (DFT) was originally introduced as a computationally feasible model for the electronic structure of small molecules and simple periodic crystals. 50 years on, thanks to advances in computing power, it is routinely used to simulate macromolecules with thousands of atoms and complex materials; it serves as a more accurate replacement of empirical interatomic forces in computatioanl molecular mechanics (defects; fracture) and molecular dynamics. In my talk I survey, from a mathematical point of view, some of the fascinating variational problems and multiscale phenomena related to DFT. Some aspects have recently been understood, at least in model systems (e.g. the dilute and concentrated limits of exact Hohenberg-Kohn DFT, see H.Chen/G.F., Multiscale Modeling Simul., 2015; these lead into Hartree-Fock theory and, surprisingly, optimal transport). Other problems remain widely open (e.g. existence of dislocations under DFT forces, or understanding the Mermin free energy at finite temperature).
► Dmitry Golovaty
Title: Dimension Reduction for the Landau-de Gennes Model Describing Thin Nematic Films
Abstract: I will present a recent $\Gamma$-convergence result that describes the behavior of the Landau-de Gennes (LdG) model for a nematic liquid crystalline film in the limit of vanishing thickness. The film is assumed to be attached to a fixed surface. In the LdG theory, an equilibrium liquid crystal configuration is described by a tensor-valued order parameter field - a nematic Q-tensor - that minimizes an energy consisting of the bulk potential, elastic, and surface (weak anchoring) energy contributions. In the asymptotic regime of vanishing thickness, the anchoring energy plays a greater role and it is essential to understand its influence on the structure of the minimizers of the derived limiting surface energy. I will assume general weak anchoring conditions that are chosen so as to enforce that a anchoring-energy-minimizing nematic $Q$-tensor has the normal to the film as one of its eigenvectors. I will outline a general convergence result and then discuss the limiting problem in several parameter regimes.
► Richard James
Title: Twisted X-Rays, Orbital Angular Momentum and the Determination of Atomic Structure
Abstract: We find exact solutions of Maxwell's equations that are the precise analog of plane waves, but in the case that the translation group is replaced by the Abelian helical group. These waves display constructive/destructive interference with helical atomic structures, in the same way that plane waves interact with crystals. We show how the resulting far-field pattern can be used for structure determination. We test the method by doing theoretical structure determination on the Pf1 virus from the Protein Data Bank. The underlying mathematical idea is that the structure is the orbit of a group, and this group is a subgroup of the invariance group of the differential equations.
Joint work with Dominik Juestel and Gero Friesecke. (Acta Crystallographica A72 and SIAM J. Appl Math).
► Robert Kohn
Title: Prediction without Probability: A PDE Approach to some Two-Player Games from Machine Learning
Abstract: In the machine learning literature, one approach to "prediction" assumes that advice is available from a finite number of "experts". The best prediction in this setting is the one that "minimizes regret", i.e. minimizes the worst-case shortfall relative to the best performing expert. My talk discusses a particular problem of this type, which takes the form of a randomized-strategy two-player game. I'll explain how it can be addressed using ideas from optimal control and partial differential equations. The main idea is to consider a suitable continuum limit, and to characterize the value function using a nonlinear PDE. Due to the special structure of the example, the value function even has an exact formula. As a consequence, one knows exactly how the experts' guidance should be weighted to obtain an optimal result.
This is joint work with Nadejda Drenska.
► Mitchell Luskin
Title: Mathematical Modeling of Incommensurate Materials
Abstract: Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation. Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.
► Donatella Marini
Title: Serendipity Nodal VEM Spaces
Abstract: We introduce a new variant of Nodal Virtual Element spaces that mimics the "Serendipity Finite Element Methods" and allows to reduce (often in a significant way) the number of internal degrees of freedom. When applied to the faces of a three-dimensional decomposition, this allows a reduction in the number of degrees of freedom: an improvement that cannot be achieved by a simple static condensation. On triangular and tetrahedral decompositions the new elements (contrary to the original VEMs) reduce exactly to the classical Lagrange FEM. On quadrilaterals and hexahedra the new elements are quite similar (and have the same amount of degrees of freedom) to the Serendipity Finite Elements, but are much more robust with respect to element distortions. On more general polytopes the Serendipity VEMs are the natural (and simple) generalization of the simplicial case.
► Hiroshi Matano
Title: Front Propagation in an Anisotropic Allen-Cahn Equation
Abstract: In this talk we discuss the long time behavior of spreading fronts in anisotropic Allen-Cahn type nonlinear diffusion equations on ${\mathbb R}^N\ (N\geq 2)$. Here, the term {\it spreading fronts} roughly means the level surfaces of solutions that expand toward infinity from a localized state. Among other things we show that the spreading level surfaces become smooth surfaces in finite time and that their shape is well approximated by the Wulff shape associated with the anisotropy of the equation as time tends to infinity. Furthermore, the solution profile near the front converges locally to a traveling wave profile in the $C^2$ sense in every direction of expansion. We obtain these results by constructing a fine pair of super- and subsolutions and by using a certain Liouville type theorem for entire solutions of the anisotropic Allen-Cahn equation satisfying certain conditions at infinity.
This is joint work with Yoichiro Mori and Mitsunori Nara.
► Léonard Monsaingeon
Title: A JKO Splitting Scheme for Kantorovich-Fisher-Rao Gradient Flows
Abstract: In this talk I will present a variant of the Jordan-Kinderlehrer-Otto scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. The splitting scheme successively performs a time step for the quadratic Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao distance. Exploiting some inf-convolution structure of the new metric we show convergence of the process for the standard class of energy functionals under suitable structural assumptions. On the one hand we prove existence of weak solutions for a whole class of reaction-advection-diffusion equations, and on the other hand this process is constructive and well adapted to available numerical solvers.
This is joint work with Thomas Gallouët.
► Felix Otto
Title: Convergence of the Thresholding Scheme for Multi-Phase Mean Curvature Flow
Abstract: We consider the thresholding scheme, a time discretization for mean curvature flow introduced by Bence-Merriman-Osher; and prove a convergence result in the multi-phase case. The result establishes convergence towards a weak formulation in the framework of sets of finite perimeter. Multi-phase mean-curvature flow is a model for grain growth, especially when one allows for surface tensions that depend on the pairs of grains, as we do. The proof is based on the interpretation of the thresholding scheme as a minimizing movement scheme, which means that the thresholding scheme preserves the structure of (multi-phase) mean curvature flow as a gradient flow w.\ r.\ t.\ the total interfacial energy. More precisely, the thresholding scheme is a minimizing movement scheme for an energy functional that $\Gamma$-converges to the total interfacial energy (joint work with Selim Esedoglu). Our proof is similar in spirit to the convergence results of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker of another minimizing movement scheme for mean curvature flow. In particular, ours is a conditional convergence result, in the sense that we assume that the energy of the approximation converges to the energy of the limit. In addition, we appeal to an argument of De Giorgi to show that the limit satisfies the gradient flow structure in the sense of Brakke.
This is joint work with Tim Laux.
► Russell Schwab
Title: A Min-Man Formula for Lipschitz Operators that Satisfy the Global Comparison Principle
Abstract: We investigate Lipschitz maps, I, mapping $C^2(D) \to C(D)$, where $D$ is an appropriate domain. The global comparison principle (GCP) simply states that whenever two functions are ordered in D and touch at a point, i.e. $u(x)\leq v(x)$ for all $x$ and $u(z)=v(z)$ for some $z \in D$, then also the mapping I has the same order, i.e. $I(u,z)\leq I(v,z)$. It has been known since the 1960's, by Courrège, that if I is a linear mapping with the GCP, then I must be represented as a linear drift-jump-diffusion operator that may have both local and integro-differential parts. It has also long been known and utilized that when I is both local and Lipschitz it will be a min-min over linear and local drift-diffusion operators, with zero nonlocal part. In this talk we discuss some recent work that bridges the gap between these situations to cover the nonlinear and nonlocal setting for the map, I. These results open up the possibility to study Dirichlet-to-Neumann mappings for fully nonlinear equations as integro-differential operators on the boundary.
This is joint work with Nestor Guillen.
► Marshall Slemrod with Amit Acharya, Gui-Qiang Chen, and Dehua Wang
Title: Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions
Abstract: We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the interconnected nonlinear partial differential equations. In this paper we develop such connections for the case of two spatial dimensions and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the Gauss-Codazzi equations through examples for the Euler equations for fluids and the Euler-Lagrange equations for elastic solids. These results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics.
► Xiang Xu
Title: A stable Scheme for a 2D Dynamic Q-tensor Model
Abstract: We propose an unconditionally stable scheme for a 2D dynamic Q-tensor model describing nematic liquid crystals. This dynamic Qtensor model is considered as a gradient flow generated by the liquid crystal free energy that contains a cubic term, which makes the free energy unbounded from below. By using a stabilizing technique, we obtain an unconditionally stable scheme and establish the unique solvability and convergence of this scheme, which also leads to the well-posedness of the original PDE system.

This conference is supported by the National Science Foundation under Grant No. DMS-1601475, by the Center for Nonlinear Analysis and by the Department of Mathematical Sciences at Carnegie Mellon University.

NSF    Center for Nonlinear Analysis