Scientific Program

June, 20-24

Schedule

	Mon	Tue	Wed	Thu	Fri
9:30-10:15	Shagholian	Bardi	Rossi	Martel	Bernard
10:30-11:15	Valdinoci	Patrizi	Terrone	Oliveira	Cagnetti
11:30-12:15	Oberman	Urbano	Tudorascu	Camilli	Fathi
12:15-14:30	Lunch	Lunch	Lunch	Lunch	Lunch
14:30-15:15	Gualdani	Perez		Siconolfi
15:30-16:15	Rifford	Santambrogio		Pavlovic

Speakers and Abstracts

• Martino Bardi

Critical value of some non-convex Hamiltonians

We consider Hamiltonians H convex in the first n moment variables and concave in the other m, e.g., the difference of two usual convex and coercive Hamiltonians, the former in n and the latter in m variables. At least formally, one can associate to it a variational problem where one seeks trajectories whose first n components minimize and the last m maximize a convex-concave Lagrangian. We show that the theory of differential games allows to make this link rigorous, at least in part. We also define as critical value of H the infimum of the constants c such that the stationary Hamilton-Jacobi equation with right hand side c has a (viscosity) subsolution. We prove that some of the properties of the critical value well-known in the convex case still hold for a large class of non-convex Hamiltonians. Finally we prove the existence and a formula for the critical value of some convex-concave Hamiltonians.

• Patrick Bernard
• Filippo Cagnetti

Adjoint methods for Obstacle problems and Weakly coupled systems of PDE

The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton--Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of common approximation procedures are derived. This is a joint work with Diogo Gomes and Hung V. Tran.

• Fabio Camilli

Large time behavior of systems of Hamilton-Jacobi equations

The aim of this talk is to discuss the large time behavior of solutions to weakly coupled systems of Hamilton-Jacobi equations. I will present a result which can be seen as the analogous of the theorem of Namah-Roquejoffre for the case of systems.The proof is based on new comparison, existence and regularity results for systems. (joint work with Olivier Ley, P. Loreti and V. Nguyen)

• Albert Fathi

Convergence of discounted solutions of the Hamilton-Jacobi equation

This is a joint work with Renato Iturriaga.

We consider a Hamiltonian $H:\R^n\times \R^n\to\R, (x,p)\mapsto H(x,p)$ that is $\Z^n$ periodic in the first variable $x$, and convex superlinear in the second variable $p$.

For $\lambda>0$ we consider (viscosity) solutions of the discounted Hamilton-Jacobi equation $$\lambda u_\lambda+ H(x,Du_\lambda (x))=c[0],$$ where $c[0]$ is the unique constant $c$ such that the stationary Hamilton-Jacobi equation $$H(x,Du(x))=c$$ has a viscosity solution. It s well-known that $u_\lambda$ is unique and that $u_\lambda$ accumulates on viscosity solutions of the stationary Hamilton-Jacobi equation, when $\lambda \to 0$.

We address the problem of actual convergence of $u_\lambda$ when $\lambda\to 0$.

Using weak KAM theory, we can show that $u_\lambda$ converges to a unique viscosity solution of the stationary Hamilton-Jacobi equation, provided the Mather quotient of the Aubry Mather set has measure $0$ for the 1-dimensional Hausdorff measure.

In particular, this is the case when the Mather quotient of the Aubry set is at most countable (generic condition on $H$), and also when $n\leq 3$ and $H$ is C$^4$.

We will recall the elements from Aubry-Mather and weak KAM theory that are necessary to understand the lecture.

• Maria Pia Gualdani

A factorization method for non-symmetric linear operator: enlargement of the functional space while preserving hypo-coercivity.

We present a factorization method for non-symmetric linear operators: the method allows to enlarge functional spaces while preserving spectral properties for the considered operators. In particular, spectral gap and related convergence towards equilibrium follow easily by hypo-coercivity and resolvent estimates. Applications of this theory on several kinetic equations will be presented.

• Yvan Martel

Inelastic interaction of solitons for the quartic gKdV equation

We present recent works in collaboration with Frank Merle concerning the interaction of two solitons for the quartic (gKdV) equation. In two specific asymptotic cases (almost equal speeds / very different speeds), we can describe the collision into details. In particular, we prove that at the main order, the two solitons are preserved by the interaction as for the original integrable KdV equation. However, unlike for the KdV equation, we prove that the collision is inelastic, thus showing the non integrable nature of the quartic generalized KdV equation.

References:

• Y. Martel et F. Merle, Description of two soliton collision for the quartic gKdV equation, to appear in Annals of Math. http://arxiv.org/abs/0709.2672
• Y. Martel et F. Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation, to appear in Inventiones Mathematicae. http://arxiv.org/abs/0910.3204
• Adam Oberman

Direct solvers for the Monge-Ampère equation with Optimal Transportation boundary conditions

The Optimal Transportation problem with quadratic cost can be solved via the elliptic Monge-Ampère Partial Differential Equation (PDE) with nonlocal boundary conditions. Up until now, building numerical solutions for the Monge-Ampère PDE has been a challenge, even with Dirichlet boundary conditions.

Indirect methods, such at the fluids reformulation by Brenier-Benamou, as possible, but a direct solver is preferable. Recently several groups of researchers have proposed numerical schemes. Unfortunately these schemes fail to converge, or converge only in the case of smooth solutions. I'll show how naive schemes can work well for smooth solutions, but break down in the singular case. This makes having a convergent scheme even more important.

Starting with the Dirichlet problem, I will present a finite difference scheme which is the only scheme proven to converge to weak (viscosity) solutions. Building on the original discretization, I'll describe modifications which improve the accuracy and solution speed. Finally, I will show how to solve the problem with Optimal Transportation boundary conditions.

This is joint work with my PhD student, Brittany Froese.

• Filipe Oliveira

Local and Global well-posedness for the critical Schrodinger-Debye System

We will begin this talk by presenting a small survey concerning the existence of local and global solutions to the $d$-dimensional Cubic Nonlinear Schrodinger Equation. We will then establish local well-posedness results for the IVP associated to the Schrodinger-Debye (SD) system in dimensions $d=2,3.$ Finally, in the critical case $(d=2)$, we will show a rather unexpected property of the SD system: the absence of blow-up for initial data in the energy space $H^1\times L^2$.

This is a joint work with Jorge D. Silva (IST-UTL) and Adán Corcho (UFRJ).

• Natasa Pavlovic

On the Cauchy problem for Gross-Pitaevskii hierarchies

The Gross-Pitaevskii (GP) hierarchy is an infinite system of coupled linear non-homogeneous PDEs, which appear in the derivation of the nonlinear Schr\"{o}dinger equation (NLS). Inspired by the PDE techniques that have turned out to be useful on the level of the NLS, we realized that, in some instances we can introduce analogous techniques at the level of the GP. In this talk we will discuss some of those techniques which we use to study well-posedness for GP hierarchies. The talk is based on joint work with Thomas Chen.

• Stefania Patrizi

Homogenization of integro-differential equations describing dislocation dynamics and applications

We will present a result concerning the homogenization of an integro-differential equation describing dislocation dynamics. Our model involves both an anisotropic Lévy operator of order 1 and a potential depending periodically on $u/\epsilon$. The limit equation is a non-local Hamilton-Jacobi equation, which is an effective plastic law for densities of dislocations moving in a single slip plane. In homogenization theory, the so called effective Hamiltonian in the limit equation is usually unknown. We will explain how to determine its asymptotic behavior at the origin, in the special case when the non-local operator is the fractional Laplacian in dimension 1. The recovered plastic law is known in physics as the Orowan's law.

This is a joint research with Régis Monneau (ENPC, Paris).

• Mayte Pérez-Llanos

A homogenization process for the strong $p(x)$-Laplacian

In this talk we analyze a homogenization process that takes place in a problem involving the strong $p(x)-$Laplacian. Precisely, we consider the following problem, closely related with Tug-of-War games

$$ \left\{\begin{array}{ll} -a(x/\ve)\Delta_\infty u(x) -b(x/\ve)\Delta u(x)=0, \quad &x\in \Omega,\\ u(x)=g(x),& x\in \partial\Omega, \end{array}\right. $$

with $a,b$ continuous functions such that $a+b=1$ and $b>0$ for every $x\in \Omega$, being $\ve$ the homogenization parameter. Classical results in regularity theory yield the convergence of at least a subsequence of $u_{\ve}$ to some continuous function, as $\ve\to 0$. We determine the equation verified by this limit, as well as some of its properties.

• Ludovic Rifford

Mass Transportation on Surfaces

We address the problem of regularity of optimal transport maps on surfaces. We give necessary and sufficient conditions for the so-called Transport Continuity Property and discuss a list of examples.

• Julio Rossi

On the best Lipschitz extension problem for a discrete distance and the discrete &\infty$-Laplacian.

This talk is concerned with the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete $\infty$-Laplacian problem, which arise as the dynamic programming formula for the value function of some $\varepsilon$-tug-of-war games. As in the classical case, we obtain the absolutely minimizing Lipschitz extension of a datum $f$ by taking the limit as $p\to \infty$ in a nonlocal $p$--Laplacian problem.

Joint work with J Mazon and J Toledo.

• Antonio Siconolfi

Homogenization of geometrical Hamilton-Jacobi equations and models of turbulent combustion

We recover some recent results of Cardaliaguet-Nolen-Souganidis about the periodic homogenization of the G-equation, by adopting a different approach based on metric techniques.

The level-set method has been introduced in the modelization of turbulent combustion, in the form of the so-called G-equation, since a long time. However, only recently, some subtleties in the mathematical analysis of the model have received attention. The appealing point in this respect is that the G-equation looses coercivity when the turbulent velocity exceeds the laminar one. This implies, as pointed out by Cardaliaguet-Nolen-Souganidis and Xin-Yu, that some condition must be imposed on the divergence of the vector field representing the turbulence to get the homogenization, in the periodic setting.

As already said, we look at the matter from a metric angle. It is well known, at least in the coercive case, that to any positively homogeneous Hamiltonian can be associated a path distance, which is obtained through minimization of an intrinsic length functional. The metric counterpart of the lack of coercivity is that such distance can become infinite. We are then interested to the paths possessing finite intrinsic length, and it turns out that they are integral curves of a certain cone dynamics, which actually is the main object of our investigation. Under suitable assumptions, we show a sort of ergodicity property for such dynamics, when restricted to the torus. This is, in our approach, the key step for proving the homogenization result.

• Henrik Shagholian

Quadrature Domains and related topics

A quadrature domain (QD) is a domain that admits a generalized mean value property for harmonic functions

\int_D h dx = \int h d\mu

for a given measure $\mu$, and all integrable harmonic functions $h$ in $D$. The appropriate ball is an example of a QD with respect to the Dirac mass.

This theory was developed systematically by P.J. Davis in 60's and later rediscovered by Aharanov-Shapiro in early 70's, and has since then been under intensive investigation. The theory has found a tremendous numbers of applications in various areas: Complex function theory (Schwarz function), Hele-Shaw flow of viscous fluid, Obstacle type free boundaries, Laplacian growth, Internal Diffusion limited aggregation, Random Matrices, Quantum hall theory, Operator Theory, .

The connection between a quadrature domain and obstacle type free boundary can be easily seen if in the above identity we replace $h$ by the fundamental solution $F(x,y)$ of the Laplace operator. Indeed letting

u(x) = \Int_D F(x,y) dy - \int F(x,y) d\mu_y

we shall have

\Delta u = \chi_{\{ D \}} - \mu,

and $u=0$ in $D^c$. Here $F$ is a normalized Fundamental solution. This somehow resembles the well-known obstacle type free boundary.

In this talk I shall discuss this problem and some recent applications as well as generalizations into two and multi phases.

• Filippo Santabrogio

Very degenerate elliptic equations and applications

We'll discuss about variational nonlinear equations of the form $\nabla\cdot G(\nabla u) = f$, where $G=\nabla H$ is the gradient of a convex function identically vanishing on a ball around the origin. They are similar to a $p-$Laplacian equation but much more degenerate. We will see that they appear both from relaxation of non-convex problems in material sciences and in modelization of traffic congestion (and the degeneracy is the natural assumption in such a framework) and give some regularity results on $G(\nabla u)$ (Sobolev and continuuity; no direct result may be expected on $u$ itself), motivated by applications. The links with optimal transport, with the minimal flow problem by Beckmann and with trasport densities will be presented as well.

• Gabriele Terrone

Homogenization of some non-coercive Hamilton-Jacobi Equations.

The coercivity in the gradient variable of the Hamiltonian plays a central role in the known theory of Homogenization of Hamilton-Jacobi Equations. Not much is known for non-coercive Hamiltonians, and the development of a sufficiently general theory faces the obstruction of several counterexamples for pointwise convergence.

The results I will discuss in this talk focuses mainly on min-max Hamiltonians arising in deterministic optimal control and differential games, and are based on a game-theoretical interpretation of the problem. I will first describe classes of equations for which homogenization fails. Then I will present some homogenization result under partial coercivity and some restriction on the oscillating variables. Finally, in a two-dimensional example, I will compute explicitely the effective equation and the effective differential game.

This research is a joint work with Martino Bardi.

• Adrian Tudorascu

A Weak KAM theorem in the space of probabilities on the torus (Joint work with W. Gangbo)

We consider the compact metric space of probabilities on the torus endowed with the periodic quadratic Wasserstein distance. We use a finite-to-infinite dimension approach to prove a Weak KAM theorem for a ``mechanical'' Lagrangian with an interaction potential with certain symmetry.

• José Miguel Urbano

p(x)-Harmonic functions with unbounded exponent in a subdomain

We study the Dirichlet problem for the p(x)-Laplacian, when the variable exponent p(x) is infinite in a subdomain D of the reference domain U. The main issue is to give a proper sense to what a solution is and we consider the limit of the solutions u_n corresponding to the problem obtained by replacing p(x) with p_n(x) = min (p(x),n). Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, infinity-harmonic within D. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of U.

This is a joint work with Juan J. Manfredi and Julio D. Rossi.

• Enrico Valdinoci

Phase transitions and minimal surfaces in (non)local setting

We discuss some regularity, rigidity and symmetry properties for solutions of nonlocal phase transitions, in connection with their local or nonlocal limit interface.