# Seminar Schedule Fall 2011

date speaker title host(s)
Oct 3 Takis Souganidis (Chicago)
Stochastic homogenization of the G-equation

Armstrong
Partial regularity for fully nonlinear elliptic equations

Local speaker
Oct 17 Russell Schwab (Carnegie Mellon)
On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)

Armstrong
October 24 ( with Geometry/Topology seminar) Valentin Ovsienko (University of Lyon) Marí Beffa
Oct 31 Adrian Tudorascu (West Virginia University)
Weak Lagrangian solutions for the Semi-Geostrophic system in physical space

Feldman
Nov 7 James Nolen (Duke) Armstrong
Nov 21 (Joint with Analysis seminar) Betsy Stovall (UCLA) Seeger
Dec 5 Charles Smart (MIT)
Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian

Armstrong

# Seminar Schedule Spring 2011

date speaker title host(s)
Jan 24 Bing Wang (Princeton)
The Kaehler Ricci flow on Fano manifold

Viaclovsky
Mar 15 (TUESDAY) at 4pm in B139 (joint wit Analysis) Francois Hamel (Marseille)
Optimization of eigenvalues of non-symmetric elliptic operators

Zlatos
Mar 28 Juraj Foldes (Vanderbilt)
Symmetry properties of parabolic problems and their applications

Zlatos
Apr 11 Alexey Cheskidov (UIC)
Navier-Stokes and Euler equations: a unified approach to the problem of blow-up

Kiselev
Date TBA Mikhail Feldman (UW Madison) TBA Local speaker
Date TBA Sigurd Angenent (UW Madison) TBA Local speaker

# Seminar Schedule Fall 2010

date speaker title host(s)
Sept 13 Fausto Ferrari (Bologna) Feldman
Sept 27 Arshak Petrosyan (Purdue) Feldman
Oct 7, Thursday, 4:30 pm, Room: 901 Van Vleck. Special day, time & room. Changyou Wang (U. of Kentucky) Feldman
Oct 11 Philippe LeFloch (Paris VI) Feldman
Oct 29 Friday 2:30pm, Room: B115 Van Vleck. Special day, time & room. Irina Mitrea (IMA) WiMaW
Nov 1 Panagiota Daskalopoulos (Columbia U) Feldman
Nov 8 Maria Gualdani (UT Austin) Feldman
Nov 18 Thursday 1:20pm Room: 901 Van Vleck Special day & time. Hiroshi Matano (Tokyo University) Angenent & Rabinowitz
Nov 29 Ian Tice (Brown University) Feldman
Dec. 8 Wed 2:25pm, Room: 901 Van Vleck. Special day, time & room. Hoai Minh Nguyen (NYU-Courant Institute) Feldman

# Abstracts

### Fausto Ferrari (Bologna)

Semilinear PDEs and some symmetry properties of stable solutions

I will deal with stable solutions of semilinear elliptic PDE's and some of their symmetry's properties. Moreover, I will introduce some weighted Poincaré inequalities obtained by combining the notion of stable solution with the definition of weak solution.

### Arshak Petrosyan (Purdue)

Nonuniqueness in a free boundary problem from combustion

We consider a parabolic free boundary problem with a fixed gradient condition which serves as a simplified model for the propagation of premixed equidiffusional flames. We give a rigorous justification of an example due to J.L. V ́azquez that the initial data in the form of two circular humps leads to the nonuniqueness of limit solutions if the supports of the humps touch at the time of their maximal expansion.

This is a joint work with Aaron Yip.

### Changyou Wang (U. of Kentucky)

Phase transition for higher dimensional wells

For a potential function $\displaystyle{ F }$ that has two global minimum sets consisting of two compact connected Riemannian submanifolds in $\displaystyle{ \mathbb{R}^k }$, we consider the singular perturbation problem:

Minimizing $\displaystyle{ \int \left(|\nabla u|^2+\frac{1}{\epsilon^2} F(u)\right) }$ under given Dirichlet boundary data.

I will discuss a recent joint work with F.H.Lin and X.B.Pan on the asymptotic, as the parameter $\displaystyle{ \epsilon }$ tends to zero, in terms of the area of minimal hypersurface interfaces, the minimal connecting energy, and the energy of minimizing harmonic maps into the phase manifolds under both Dirichlet and partially free boundary data. Our results in particular addressed the static case of the so-called Keller-Rubinstein-Sternberg problem.

### Philippe LeFloch (Paris VI)

Kinetic relations for undercompressive shock waves and propagating phase boundaries

I will discuss the existence and properties of shock wave solutions to nonlinear hyperbolic systems that are small-scale dependent and, especially, contain undercompressive shock waves or propagating phase boundaries. Regularization-sensitive patterns often arise in continuum physics, especially in complex fluid flows. The so-called kinetic relation is introduced to characterize the correct dynamics of these nonclassical waves, and is tied to a higher-order regularization induced by a more complete model that takes into account additional small-scale physics. In the present lecture, I will especially explain the techniques of Riemann problems, Glimm-type scheme, and total variation functionals adapted to nonclassical shock waves.

### Irina Mitrea

Boundary Value Problems for Higher Order Differential Operators

As is well known, many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator L in a domain D.

When L is a differential operator of second order a variety of tools are available for dealing with such problems including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. The situation when the differential operator has higher order (as is the case for instance with anisotropic plate bending when one deals with fourth order) stands in sharp contrast with this as only fewer options could be successfully implemented. Alberto Calderon, one of the founders of the modern theory of Singular Integral Operators, has advocated in the seventies the use of layer potentials for the treatment of higher order elliptic boundary value problems. While the layer potential method has proved to be tremendously successful in the treatment of second order problems, this approach is insufficiently developed to deal with the intricacies of the theory of higher order operators. In fact, it is largely absent from the literature dealing with such problems.

In this talk I will discuss recent progress in developing a multiple layer potential approach for the treatment of boundary value problems associated with higher order elliptic differential operators. This is done in a very general class of domains which is in the nature of best possible from the point of view of geometric measure theory.

Ancient solutions to geometric flows

We will discuss the clasification of ancient solutions to nonlinear geometric flows. It is well known that ancient solutions appear as blow up limits at a finite time singularity of the flow. Special emphasis will be given to the 2-dimensional Ricci flow. In this case we will show that ancient compact solution is either the Einstein (trivial) or one of the King-Rosenau solutions.

### Maria Gualdani (UT Austin)

A nonlinear diffusion model in mean-field games

We present an overview of mean-field games theory and show recent results on a free boundary value problem, which models price formation dynamics. In such model, the price is formed through a game among infinite number of agents. Existence and regularity results, as well as linear stability, will be shown.

### Hiroshi Matano (Tokyo University)

Traveling waves in a sawtoothed cylinder and their homogenization limit

My talk is concerned with a curvature-dependent motion of plane curves in a two-dimensional cylinder with spatially undulating boundary. In other words, the boundary has many bumps and we assume that the bumps are aligned in a spatially recurrent manner.

The goal is to study how the average speed of the traveling wave depends on the geometry of the domain boundary. More specifically, we consider the homogenization problem as the boundary undulation becomes finer and finer, and determine the homogenization limit of the average speed and the limit profile of the traveling waves. Quite surprisingly, this homogenized speed depends only on the maximal opening angles of the domain boundary and no other geometrical features are relevant.

Next we consider the special case where the boundary undulation is quasi-periodic with m independent frequencies. We show that the rate of convergence to the homogenization limit depends on this number m.

This is joint work with Bendong Lou and Ken-Ichi Nakamura.

### Ian Tice (Brown University)

Global well-posedness and decay for the viscous surface wave problem without surface tension

We study the incompressible, gravity-driven Navier-Stokes equations in three dimensional domains with free upper boundaries and fixed lower boundaries, in both the horizontally periodic and non-periodic settings. The effect of surface tension is not included. We employ a novel two-tier nonlinear energy method that couples the boundedness of certain high-regularity norms to the algebraic decay of lower-regularity norms. The algebraic decay allows us to balance the growth of the highest order derivatives of the free surface function, which then allows us to derive a priori estimates for solutions. We then prove local well-posedness in our energy space, which yields global well-posedness and decay. The novel LWP theory is established through the study of the linear Stokes problem in moving domains. This is joint work with Yan Guo.

### Hoai Minh Nguyen (NYU-Courant Institute)

Cloaking via change of variables for the Helmholtz equation

A region of space is cloaked for a class of measurements if observers are not only unaware of its contents, but also unaware of the presence of the cloak using such measurements. One approach to cloaking is the change of variables scheme introduced by Greenleaf, Lassas, and Uhlmann for electrical impedance tomography and by Pendry, Schurig, and Smith for the Maxwell equation. They used a singular change of variables which blows up a point into the cloaked region. To avoid this singularity, various regularized schemes have been proposed. In this talk I present results related to cloaking via change of variables for the Helmholtz equation using the natural regularized scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the authors used a transformation which blows up a small ball instead of a point into the cloaked region. I will discuss the degree of invisibility for a finite range or the full range of frequencies, and the possibility of achieving perfect cloaking. If time permits, I will mention some results related to the wave equation.

### Bing Wang (Princeton)

The Kaehler Ricci flow on Fano manifold

We show the convergence of the Kaehler Ricci flow on every 2-dimensional Fano manifold which admits big $\displaystyle{ \alpha_{\nu, 1} }$ or $\displaystyle{ \alpha_{\nu, 2} }$ (Tian's invariants). Our method also works for 2-dimensional Fano orbifolds. Since Tian's invariants can be calculated by algebraic geometry method, our convergence theorem implies that one can find new Kaehler Einstein metrics on orbifolds by calculating Tian's invariants. An essential part of the proof is to confirm the Hamilton-Tian conjecture in complex dimension 2.

### Francois Hamel (Marseille)

Optimization of eigenvalues of non-symmetric elliptic operators

The talk is concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of $\displaystyle{ R^n }$. To each operator in a given domain, we can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.

### Juraj Foldes (Vanderbilt)

Symmetry properties of parabolic problems and their applications

Positive solutions of nonlinear parabolic problems can have a very complex behavior. However, assuming certain symmetry conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is 'stable'; more specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. As an application, we show new results on convergence of solutions to a single equilibrium.

### Alexey Cheskidov (UIC)

Navier-Stokes and Euler equations: a unified approach to the problem of blow-up

The problems of blow-up for Navier-Stokes and Euler equations have been extensively studied for decades using different techniques. Motivated by Kolmogorov's theory of turbulence, we present a new unified approach to the blow-up problem for the equations of incompressible fluid motion. In particular, we present a new regularity criterion which is weaker than the Beale-Kato-Majda condition in the inviscid case, and weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case.

### Takis Souganidis (Chicago)

Stochastic homogenization of the G-equation

The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.

Partial regularity for fully nonlinear elliptic equations

I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.

### Russell Schwab (Carnegie Mellon)

On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)

Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])

### Valentin Ovsienko (University of Lyon)

The pentagram map and generalized friezes of Coxeter

The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map. In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.

### Adrian Tudorascu (West Virginia University)

Weak Lagrangian solutions for the Semi-Geostrophic system in physical space

Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. In spite of significant progress achieved in the analysis of the SG in dual space (i.e. the system obtained from the SG by a special change of variables), there are no existence results on the SG in physical space. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. We have proved existence of such solutions in the case of discrete measures in dual space. The talk is based on joint work with M. Feldman.

### James Nolen (Duke)

Normal approximation for a random elliptic PDE

I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. This talk is about the energy dissipation rate, which is a quadratic functional of the solution and a bulk property of the random material. For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance. I'll give an error estimate for this approximation in total variation.

### Betsy Stovall (UCLA)

We will discuss recent work concerning the cubic Klein--Gordon equation u_{tt} - \Delta u + u \pm u^3 = 0 in two space dimensions with real valued initial data in the energy space, u(0) in H^1, u_t(0) in L^2. We show that in the defocusing case, solutions are global and have finite L^4 norm (in space and time). In the focusing case, we characterize the dichotomy between such behavior and finite time blowup for initial data having energy less than that of the ground state. In this talk, we will pay particular attention to connections with certain questions arising in harmonic analysis.

This is joint work with Rowan Killip and Monica Visan.

### Charles Smart (MIT)

Optimal Lipschitz Extensions and Regularity for the Infinity Laplacian

A classical theorem of Kirszbraun states that any Lipschitz function $f : A \to \R^m$ defined on a set $A \subseteq \R^n$ can be extended to all of $\R^n$ without increasing the Lipschitz constant. The search for a canonical such extension leads one to the notion of optimal Lipschitz extension. I will discuss joint work with Evans on the regularity of optimal Lipschitz extensions in the scalar $m = 1$ case and joint work with Sheffield on the vector valued $m > 1$ case.