Titles and Abstracts

Title: Stochastic (non)-homogenization of some non-convex Hamilton-Jacobi equations
Abstract: I will discuss some progress about the homogenization of non-convex Hamilton-Jacobi equations in random media. I will revisit the recent counter-example of Ziliotto who constructed a coercive but non-convex Hamilton-Jacobi equation with stationary ergodic random potential field for which homogenization does not hold. We have extended this result showing that for any Hamiltonian with a strict saddle-point there is a random stationary ergodic potential field V so that homogenization does not hold for the Hamiltonian $H = h(p)-V(x)$. If there is time I will also discuss a positive result, under a finite range of dependence assumption we show that homogenization holds for Hamiltonians with strictly star-shaped sub-level sets. Talk is based on joint work with P. Souganidis.

Title: Partial regularity for doubly nonlinear parabolic systems
Abstract: Various models for phase transitions involve flows called doubly nonlinear evolutions. Typically, the time derivative of such flows appears in a nonlinear fashion. We study a doubly nonlinear evolution related to parabolic systems and investigate when solutions are partially regular.

Title: Regularity of solutions in semilinear elliptic theory
Abstract: We present regularity results for the classical semilinear elliptic equation $\Delta u = f(x,u)$ in $B_1$ and discuss several associated free boundary problems. The techniques are self-contained and involve an $L^2$ projection operator. Based on joint work with Minne and Nurbekyan.

Title: Slow motion for the mass-preserving Allen--Cahn equation
Abstract: We present some recent results on the asymptotic development by Gamma convergence of order two of the Cahn–Hilliard functional and applications to slow motion of solutions to the Allen-Cahn and Cahn-Hilliard equations.

Title: Elasticity with residual stress: curvature constraints at low regularity
Abstract: This lecture is concerned with the analysis of thin elastic films exhibiting residual stress at free equilibria, that can be studied through a variational model minimizing the distance of a deformation from being an isometric immersion of a given Riemannian metric. In this context, analysis of scaling of the energy minimizers in terms of the film's thickness leads to the rigorous derivation of a hierarchy of limiting theories, differentiated by the embeddability properties of the target metrics and, a-posteriori, by the emergence of isometry constraints with low regularity. This leads to questions of rigidity and flexibility of solutions to the weak formulations of the related PDEs, including the Monge-Ampere equation. In particular, we observe that the set of $C^{1,\alpha}$ solutions to the Monge-Ampere equation is dense in $C^0$ provided that $\alpha<1/7$, whereas rigidity holds when $\alpha>2/3$.

Title: Optimal Error Estimates in the Stochastic Homogenization for Elliptic Equations in Nondivergence Form
Abstract: I will present quantitative error estimates in the stochastic homogenization for uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the environment, I will identify the typical (optimal) size of the fluctuations with stretched exponential-type bounds in probability. A key ingredient of our approach is to develop a regularity theory down to microscopic scale which is essentially inherited from the homogenized equation. This talk is based on joint work with Scott Armstrong.

Title: Optimal transportation between unequal dimensions
Abstract: In this talk, we describe some recent developments concerning the Monge-Kantorovich problem of optimal transportation between probability densities on manifolds with different dimensions. We show the general case can be reduced to a nonlocal differential equation, whose solutions are not generally smooth. Under suitable topological conditions, we also prove that solutions are smooth if and only if a local variant of the same equation admits a smooth and elliptic solution. We go on to establish conditions for regularity when the target is one-dimensional.

Title: A min-max 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.

Title: Tug of war and its variants
Abstract: I will give an overview of the mathematical game of tug of war and its relationship to PDE theory. I will explain how a number of variants of tug of war have proved fruitful, while many of the more fundamental problems nonetheless remain open.

Title: The limit shape of convex hull peeling
Abstract: This is joint work with Jeff Calder. Convex peeling provides a way to generalize one dimensional order statistics to higher dimensions. We prove that, under suitable conditions, the convex peeling of a random point cloud approximates the solution of a nonlinear partial differential equation. This requires identifying a suitable scale-invariant problem and using geometry to obtain tail bounds.

Title: The threshold conjecture for geometric wave equations
Abstract: The threshold conjecture for energy critical wave equations asserts that global well-posedness and scattering holds for initial data below the ground state energy. This talk will provide an overview of recent and ongoing work on this problem, joint with Sung-Jin Oh, for the closely related models of Maxwell-Klein-Gordon and Yang-Mills.

Title: Onsager's conjecture and Kolmogorov's 5/3 law
Abstract: Motivated by Kolmogorov's theory of hydrodynamic turbulence, we consider the non-rigidity of weak solutions to the 3D incompressible homogeneous Euler equations, below a certain regularity index -- also known as the Onsager conjecture. In this talk we discuss new progress towards the resolution of the Onsager conjecture. This is joint work with T. Buckmaster and N. Masmoudi.