Schedule


All talks will take place in Math 312.
(see Mathematics building on this map)
(PDF version of the schedule)
Thursday, May 23rd
8:459:15 am
Breakfast, Cantor Lounge
9:3010:20 am
Donatella Danielli, Purdue University
The obstacle problem for the fractional heat equation: properties of
the free boundary
In this talk we will discuss the structure of the free boundary in the obstacle
problem for fractional powers of the heat operator. First introduced by M. Riesz in
1938, this nonlocal operator represents a basic model of the continuous time random
walks introduced by Montroll and Weiss. Our results are derived from the study of
a lowerdimensional obstacle problem for a class of local, but degenerate, parabolic
operators. Its analysis, in turn, hinges on the monotone character of functionals of
Almgren, Weiss, and Monneau type, and the properties of the associated blowups.
This is joint work with A. Banerjee, N. Garofalo, and A. Petrosyan.
10:3010:55 am
Coffee break, Cantor Lounge
11:0011:50 am
Ricardo Nochetto, University of Maryland
Finite Element Approximation of Fractional Diffusion
We approximate the integral formulation of the fractional
Laplacian in
bounded polyhedral domains. We discuss how to deal numerically with the
singular nonintegrable kernel, derive optimal energy estimates, and
discuss preconditioning with emphasis on the boundary behavior and how
to cope with the reduced boundary Sobolev regularity. We further examine
fractional obstacle problems and fractional minimal graphs in light of
the newly developed linear theory, with emphasis on the development of
boundary discontinuities.
2:002:50 pm
Luis Silvestre, University of Chicago
Regularity and structure of scalar conservation laws.
Scalar conservation law equations develop jump discontinuities even when the initial data is smooth. Ideally, we would expect these discontinuities to be confined to a collection of codimensionone surfaces, and the solution to be relatively smoother away from these jumps. The picture is less clear for rough initial data which is merely bounded. While a linear transport equation may have arbitrarily rough solutions, genuinely nonlinear conservation laws have a subtle regularization effect. We prove that the entropy solution will become immediately continuous outside of a codimensionone rectifiable set, that all entropy dissipation is concentrated on the closure of this set, and that the $L^\infty$ norm of the solution decays at a certain rate as t goes to infinity.
3:003:25 pm
Coffee Break, Cantor Lounge
3:304:20 pm
Felix Otto, MaxPlanck Institute
A variational approach to the regularity theory for the Monge Ampere
equation
We present a purely variational approach to the regularity theory for
the MongeAmpère equation, or rather optimal transportation,
introduced with M. Goldman.
Following the general strategy in the regularity theory of minimal surfaces,
it is based on the approximation of the displacement by a harmonic gradient,
which leads to a OneStep Improvement Lemma, and feeds into a
Campanato iteration
on the $C^{1,\alpha}$level for the displacement, capitalizing on
affine invariance.
On the one hand, this allows to reprove the $\epsilon$regularity
result (FigalliKim, De PhilippisFigalli) bypassing Caffarelli's
celebrated theory.
This also extends to boundary regularity (ChenFigalli),
which is joint work in progress with T. Miura.
On the other hand, it can be used as a
largescale regularity theory for the problem of matching the Lebesgue measure
to the Poisson measure in the thermodynamic limit.
More precisely, it can be shown that in the critical dimension two
(AmbrosioStraTrevisan), increments of the displacement are still stationary.
This is joint work with M. Goldman and M. Huesmann.
Friday, May 24th
9:009:30 am
Breakfast, Cantor Lounge
9:4510:35 am
Alessio Figalli, ETH Zürich
Generic regularity in obstacle problems
The socalled Stefan problem describes the temperature distribution in a ho
mogeneous medium undergoing a phase change, for example ice melting to water.
An important goal is to describe the structure of the interface separating the two
phases.
In its stationary version, the Stefan problem can be reduced to the classical
obstacle problem, which consists in finding the equilibrium position of an elastic
membrane whose boundary is held fixed and which is constrained to lie above a
given obstacle.
The aim of this talk is to discuss sone recent developments on the generic regu
larity of the free boundary in both problems.
10:4511:00 am
Coffee break, Cantor Lounge
11:10 am 12:00 pm
Juan Luis Vázquez, Universidad Autónoma de Madrid
Nonlocal Porous Medium and Nonlocal Thin Film Flows. Results and Open Problems
Nonlocality plays an important role in the modelling and theory of PDEs related to diffusion phenomena. The talk presents work on the existence, regularity and typical behaviour of solutions of nonlinear parabolic equations driven by fractional operators, which introduce nonlocal effects into classical settings. The models we discuss are related to porous medium and thin film equations. The connection between both equations is stressed. The problems in bounded domains offer new challenges.
1:452:35 pm
Inwon Kim, UCLA
Free boundary regularity for Porous Medium Equation with Drift
I will discuss qualitative behavior of the free boundary for
solutions of the porous medium equation with the presence of drift,
starting with a general initial data. We will discuss relevant results
for the problem with and without the drift. If the initial data has
superquadratic growth at the free boundary, then we show that the
support of the solution strictly expands relative to the streamline,
suggesting regularization. If in addition the solution is
directionally monotone in a local neighborhood, then we derive
nondegeneracy of solutions and regularity of the free boundary. This
is joint work with Yuming Zhang.
2:453:00 pm
Coffee Break, Cantor Lounge
3:104:00 pm
Sandro Salsa, Politecnico di Milano
Recent progress in free boundary problems with distributed sources:
higher regularity
We describe some recent results on twophase free boundary problems governed
by inhomogeneous elliptic equations with particular emphasis on higher regularity.
Joint work with Daniela De Silva and Fausto Ferrari.
