Category Archives: Abstracts

Quantitative results in stochastic homogenization for divergence-form equations

Scott Armstrong (Paris-Dauphine)

I will summarize some recent progress in the study of linear, divergence form elliptic equations with random coefficients. Several years ago, Gloria and Otto and collaborators used an idea from statistical mechanics (concentration of measure) to establish strong quantitative bounds on the sizes of fluctuations of solutions, and then Nolen, Mourrat and others proved central limit theorems for Dirichlet forms and obtained the scaling limits of solutions. In all of these results, concentration inequalities were the essential tool for transferring ergodic information from the coefficients (which satisfy an iid assumption) to the solutions themselves, and this requires some restrictive assumptions. In this talk, I will describe an alternative approach developed in the last several years with Smart and more recently with Kuusi and Mourrat. We obtain optimal quantitative estimates, central limit theorems, and the scaling limit of the correctors to a variant of the Gaussian Free Field– under fewer assumptions, with much stronger stochastic integrability, and without using abstract concentration of measure. Rather, our approach is to “linearize the randomness around the homogenized limit”: that is, we use renormalization arguments which reveal that certain Dirichlet forms are essentially additive quantities. The complicated nonlinear structure of the randomness is reduced to a linear one (essentially a sum of iid random variables) and a complete quantitative theory can then be easily read off.

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Nucleation and microstructure in martensitic phase transformations

John Ball (Oxford, UK)

When a new phase is nucleated in a martensitic phase transformation, it has to fit geometrically onto the parent phase. Likewise, microstructures in individual grains of a polycrystal have to fit together across grain boundaries. The talk will describe some mathematical issues involved in understanding such questions of compatibility and their influence on metastability, drawing on collaborations with C. Carstensen, P. Cesana, B. Hambly, R. D. James, K. Koumatos and H. Seiner.

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Liouville properties of fully nonlinear possibly degenerate elliptic operators and some applications.

Martino Bardi (Padova, Italy)

I will present a joint paper with Annalisa Cesaroni (Padova). We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein-Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.

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Surprise and predictability in bounded sources of non-convex balance laws

 Laura Caravenna (Padova, Italy)

In the talk I will show surprising and predictable aspects of bounded source terms in a non-convex balance law, with smooth flux,
when it admits a continuous solution. Namely, I will discuss to what extent the conservation law can be reduced to an (infinitely dimensional) system of ODEs along the characteristic curves. This correspondence is evident in the classical setting but it is surprising in this context with lack of regularity. Part of the correspondence just requires suitable definitions and smart technicality, but concerning part of it new odd unexpected behaviors show up. The presentation is mostly based on a joint work with S. Bianchini (SISSA) and G. Alberti (Pisa), and it extends previous works by several authors relative to the case of the quadratic flux.

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The master equation and the convergence problem in Mean Field Games

Pierre Cardaliaguet (Paris-Dauphine)

We will discuss the convergence,  as $N$ tends to infinity, of a system of $N$ coupled Hamilton-Jacobi equations, the Nash system. This system arises in differential game theory. We describe the limit problem in terms of the so-called “master equation”, a kind of transport equation stated on the space of probability measures. This is a just work with F. Delarue, J.-M. Lasry and P.L. Lions.

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Homogenization of a semilinear heat equation

Annalisa Cesaroni (Padova, Italy)

I will discuss the homogenization of a semilinear heat equation with vanishing viscosity and oscillating  potential depending on u/eps.  According to the rate between frequency of oscillations and vanishing factor in the viscosity, we obtain different limit behaviour of the solutions.  In the weak diffusion regime, the effective operator is discontinuous in the gradient entry, an unusual phenomenon in homogenization, and makes the analysis of the limit more challenging. Joint work with Dirr (Cardiff) and Novaga (Pisa).

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Multidimensional Shock waves and Free boundary problems for Nonlinear PDEs of Mixed Type

Gui-Qiang G. Chen (Oxford, UK)

In this talk, we will discuss some recent progress in the analysis of multidimensional shock waves and related free boundary problems for nonlinear PDEs of mixed elliptic-hyperbolic type. Further trends and open problems in this direction and their connections with some
fundamental problems in other areas will also be addressed if time permits.

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Everywhere discontinuous anisotropy of thin periodic composite plates

Mikhail Cherdantsev (Cardiff, UK)

We consider an elastic periodic composite plate in the full bending regime, i.e. when the displacement of the plate is of finite order. Both the thickness of the plate h and the period of the composite structure ε are small parameters. We start from the non-linear elasticity setting. Passing to the limit as h, ɛ→, 0 we carry out simultaneous dimension reduction and homogenisation to obtain an effective limit elastic functional which describes the asymptotic properties of the composite plate. We show, in particular, that in the regime h<< ɛ2 the limit elastic functional is discontinuously anisotropic in every direction of bending. This remarkable property (suggesting that the corresponding composite plate can be referred to as metamaterial) is due to the in-limit linearisation of the bending deformations and the multi scale interaction.

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Homogenisation in finite elasticity for composites with a high contrast in the vicinity of rigid-body motions 

Kirill Cherednichenko (Bath, UK)

I will describe a multiscale asymptotic framework for the analysis of the macroscopic behaviour of periodic two-material composites with high contrast in a finite-strain setting. I will start by introducing the nonlinear description of a composite consisting of a stiff material matrix and soft, periodically distributed inclusions. I shall then focus on the loading regimes when the applied load is small or of order one in terms of the period of the composite structure. I will show that this corresponds to the situation when the displacements on the stiff component are situated in the vicinity of a rigid-body motion. This allows to replace, in the homogenisation limit, the nonlinear material law of the stiff component by its linearised version. As a main result, I derive (rigorously in the spirit of Γ-convergence) a limit functional that allows to establish a precise two-scale expansion for minimising sequences. This is joint work with Mikhail Cherdantsev and Stefan Neukamm.

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Compensated convexity, multiscale medial axis maps, and sharp regularity of the squared distance function

Elaine Crooks (Swansea, UK)

Compensated convex transforms enjoy tight-approximation and locality properties that can be exploited to develop multiscale, parametrised methods for identifying singularities in functions. When applied to the squared distance function to a closed subset of Euclidean space, these ideas yield a new tool for locating and analyzing the medial axis of geometric objects, called the multiscale medial axis map. This consists of a parametrised family of nonnegative functions that provides a Hausdorff-stable multiscale representation of the medial axis, in particular producing a hierarchy of heights between different parts of the medial axis depending on the distance between the generating points of that part of the medial axis. Such a hierarchy enables subsets of the medial axis to be selected by simple thresholding, which tackles the well-known stability issue that small perturbations in an object can produce large variations in the corresponding medial axis. A sharp regularity result for the squared distance function is obtained as a by-product of the analysis of this multiscale medial axis map.

This is joint work with Kewei Zhang, Nottingham, and Antonio Orlando, Tucumán.

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