
Modeling the Evolution of Phase Boundaries  a Survey
This talk provides an overview of contemporary mathematical
methods to study the evolution of phase boundaries, ranging
from classical phase field models to upto date investigations of
the meso and nanoscales. Various aspects like the effect of
stress, nucleation, the role of thermodynamics, simulating the
microstructures, recent numerical improvements, and their im
portance to applications are discussed.

Miscible Flow through Porous Media
Miscible displacement is an enhanced oil recovery technique in
which a solvent, miscible with the oil, is injected into an oil
reservoir. The flow is usually convection dominated, and when
diffusion is neglected we recover the Muskat problem. The sol
vent is less viscous than the oil it is displacing and so a viscous
fingering instability is observed. We study the formation of vis
cous fingers in a miscible displacement, and show that many
thin fingers will form. Numerical simulations show that a mix
ing zone where the fingers appear. We model the growth of
this mixing zone, exploiting the large aspectratio of the fingers,
to produce a systematic derivation of the Koval model, used in
industry. The exact growth rate of the fingers is controlled by
behavior near the tips and roots, with further modeling required
in this region.

Moving Boundary Analysis of Streamer Dynamics
The minimal PDE model for a streamer discharge consists of
two reaction advectiondiffusion equations for electron or ion
densities, coupled to the Poisson equation of electrostatics. Ute
Ebert showed in her talk how a regularized moving boundary
approximation can be formulated, and she presented some solutions of this approximation.
In this talk, we present the extension of linear perturbation
theory to arbitrary ε > 0, where ε is the ratio between the
regularization length and the radius of the circle. Furthermore,
we analyze numerically the nonlinear dynamics of perturbations
of the circle solution which is the simplest uniformly translating
solution.
We also present a recent study of a periodic array of interacting streamer discharges in a strong homogeneous electric field.
Simulations of the minimal PDE model yield a phase diagram
with two regions. For large period length and/or large electric
field, the streamers branch similarly to single streamers. However, for small period length and field, the streamers reach a
uniformly translating motion and do not branch. We also show
that the unregularized moving boundary approximation of this
dynamics leads to a model identical to the one used to study
HeleShaw flow in a channel geometry. Analytical uniformly
translating solutions for this model are known since the work of
Saffman and Taylor. We show that the selected SaffmanTaylor
finger fits very well the uniformly translating finger solutions obtained from the simulations of the minimal PDE model, though
the regularization is different and the boundary approximation
breaks down at the sides of the finger.

Identification of Non Smooth Cracks by Boundary Measurements
This talk deals with the identifiability of nonsmooth defects (cracks or cavities) by boundary measurements. We prove the uniqueness of the detection by two measurements for arbitrary closed sets satisfying quasieverywhere a conductivity assumption. This is the case of a large class of closed sets, inlcuding sets with infinite number of connected components. This regularity assumption is rather to be related to the Wiener criterion than to the usual boundary smoothness.
The weak geometric stability of the detection is proved without any regularity asumption, in the frame of a finite number of defects. It allows to formally prove that a numerical approixmation by shape optimization methods converge to the defect.

Free Boundaries in Biological Aggregation Models.
This talk will discuss the appearance and simulation of free
boundary motion in biological aggregation models with nonlocal
attractive forces. Due to the attractive part clusters will form,
which however remain at finite size due to a local repulsive force
(nonlinear diffusion). Hence, a similar coarsening behavior as
in phasechange models happens, which we analyze and numerically compute based on appropriate gradient flow structures.

Integral Minimal Surfaces.
I will talk about Integral Minimal Surfaces We will discuss movement by "integral curvature" and integral minimal surfaces.
This type of problems occur when long range interactions determine the local diffusion speed along a surface.

A Chemotaxis Based Model for Concentric Patterning in the Brain.
The KellerSegel model for celltocell attraction is able to reproduce the phenomenon of critical mass: cells spatially organize themselves only if the total amount of cells is sufficiently high in some sense. A typical example is the cell density's blowup occuring when the total mass is above a certain threshold in 2D.
In this talk we briefly review the KellerSegel model and its recent extensions. As an application we suggest a 3species model that describes pattern formation arising in a variant of Multiple Sclerosis. Instead of being homogeneous plaques, the areas of damaged myeline are in fact concentric rings in Baló's Concentric Sclerosis. We show how this simple 3species model based on chemotaxis describes correctly the dynamics of this disease, and we derive several qualitative properties such as an interesting correlation between pattern formation and aggressivity of the disease.
This is joint work with H.R. Khonsari. It is closely related to recent investigations by Nadin, Perthame and Ryzhik.

Comparison of Credit Default Models
Similarities and differences in the structural (valueoffirm), copula and intensity modeling of credit default will be examined. A
complete description of the relationship of the default probability of a single firm and its default barrier will be given. We will
then outline how default correlation between two firms is naturally included in these structural models and derive the joint
default probability of the firms in this context. Comparisons
will be made with the joint default probabilities obtained from
copula and intensity models in the context of tail dependence
(independence of extreme events), calibration to firsttodefault
credit default swaps (FtD CDS) and the distribution of time
between defaults. This is joint work with students Junming
Huang, Bo Shi and Lung Kwan Tsui.

On the Existence of Flame Balls in Lean Media
I will present new existence results for solution of a reaction diffusion model of spherical flame
in lean media. I will first focus on a simplified version of the model where the Arrhenius law is
replaced by an Heavyside function with an ignition temperature and heat losses due to radiation
are modelled by a linear function of the temperature. I will present an explicit construction of the
solutions and a full description of the set of solutions. Then, I will present some existence results
concerning the full reaction diffusion system.

Free Boundary Problems with Multiple Interfaces: New Exact Solution
This talk will present new analytical solutions to a free boundary problem
in which multiple interacting interfaces are present. The mathematical problem appears
in a wide variety of physical problems (including HeleShaw flows and "streamers"
in electric fields). The new results are derived using some analytical techniques
for multiply connected planar geometries recently developed by the speaker and
are related to prior work by Tanveer [Phys. Fluids, vol 30, (1987)] on steady
bubbles in HeleShaw channels.

A TwoFluid Model for Violent Aerated Flows.
In the study of ocean wave impact on structures, one often uses Froude scaling since the dominant force is gravity. However the presence of trapped or entrained air in the water can significantly modify wave impacts. When air is entrained in water in the form of small bubbles, the acoustic properties in the water change dramatically and for example the speed of sound in the mixture is much smaller than in pure water, and even smaller than in pure air. While some work has been done to study smallamplitude disturbances in such mixtures, little work has been done on large disturbances in airwater mixtures. We propose a basic twofluid model in which both fluids share the same velocities.
It is shown that this model can successfully mimic water wave impacts on coastal structures. Even though this is a model without interface, waves can occur. Their dispersion relation is discussed and the formal limit of pure phases (interfacial waves) is considered. The governing equations are discretized by a secondorder finite volume method. Numerical results are presented. It is shown that this basic model can be used to study violent aerated flows, especially by providing fast qualitative estimates.

Interfaces in Heterogeneous Media
We consider energies which consist of an interfacial energy and
a heterogeneous (periodic or random) "bulk" or "volume" term,
with heterogeneities that change on a "small" scale. This competition of the interfacial energy and the heterogenities creates
a complex energy landscape. We present some results for the
qualitative behavior on large scales, both for the energies and
for the associated gradient flows.

Streamer Ionization Fronts As a Moving Boundary Problem
Streamer discharges are the earliest stages of sparks and lightning; they are growing cold plasma channels that propagate due
to the nonlinear coupling of electron drift, impact ionization reaction and space charge effects, which leads to a strong field
enhancement at the propagating tips of the streamer fingers.
Tens of kilometers large sprite discharges above thunderclouds
are driven by the same physical mechanisms. As streamers and
sprites propagate with velocities of 10^{5} to 10^{7} m/s, experiments
and observations only recently allow to characterize them properly.
The minimal model for a streamer discharge (e.g., in pure
gases like nitrogen or argon) consists of two reactionadvectiondiffusion equations for electron or ion densities, coupled to the
Poisson equation of electrostatics. Numerical solutions show
the formation of a thin space charge layer around the ionized
streamer interior. A moving boundary approximation for this
thin layer is of viscous fingering type. As a regularization of
the boundary dynamics, we have suggested and approximately
derived a mixed DirichletNeumann boundary condition, similar
to kinetic undercooling.
We have thoroughly analyzed the stability of convecting circles in this boundary problem, as these are the simplest uni
formly translating solution. For a particular ratio ε of regularization length over radius, the linear parturbation theory
about the convecting circle can be solved exactly, and universal
asymptotes for time to ∞ as well as the discrete spectrum can
be determined. The extension of linear perturbation theory to
arbitrary ε > 0, nonlinear solutions as well as the dynamics of
finger solutions will be treated by Fabian Brau in his talk.
The moving boundary analysis is joint work with Fabian
Brau, ChiuYen Kao, Bernard Meulenbroek, Lothar Schaer and Saleh Tanveer.

Convexity of the Optimal Stopping Boundary for the American Put Option
We show that the optimal stopping boundary for the American put
option is convex in the standard BlackScholes model. The methods
are adapted from icemelting problems and rely upon studying the
behavior of level curves of solutions to certain parabolic
differential equations.

A Geometric Approach to Overdetermined Boundary Value Problems
We use the theory of isoparametric surfaces to prove the following symmetry
result :
if the overdetermined elliptic (not necessarily uniformly elliptic) boundary value problem :
admits a weak C_{0}^{1}(Ω)solution in a connected bounded open set Ω ⊂R^{N}, then Ω must be a an euclidean ball.
The above result is proved under
minimal assumptions on both Ω and the
function
A (this is a joint work with B. Kawohl).

Recent Trends in Modelling Cancer Invasion.
The literature on cancer modelling has increased at an impressive pace during the last decade, addressing a number of different topics. Several mathematical models have been formulated
for tumors with a prescribed geometry (namely the multicellular
spheroids and the socalled tumor cords), dealing with various
aspects of their evolution (growth, metabolism, mechanical behavior, angiogenesis, treatments etc.), both in vitro and in vivo.
The possibility that such simple shapes can be unstable, giving rise to more complicated structure (even of fractal type)
has also been analyzed in a series of papers. Recently much
attention has been devoted to the critical problem of tumor invasiveness, which in turn involves the study of cells motility
(chemotaxis, haptotaxis, intra and extravasation), as well as
the ability of tumor cells to exert an aggressive action on the
host tissue. All these topics seem to be relevant to the phenomenon of metastasis. Here we want to concentrate on the
question of the acidmediated tumor invasion. Such a process
is based on the fact that the host tissue is more sensitive to pH
lowering than the tumor cells. Thus the tumor invasion can be
favored by the production of an acidic environment. The latter
is a consequence of the products of a metabolism shifted towards
glycolysis, which is typical of hypoxic tumors. Two classes of
papers will be considered. The first class studies the progression of the tumor by assuming some simple law of production
of an excess of H+ ions and contains papers dealing with onedimensional traveling waves or with spherical tumors (vascular
or not). The second class is concerned more specifically with
the metabolism of spheroids responsible for the pH lowering.
The two classes present interesting features and are in a sense
complementary, suggesting that much progress can result from
combining the two approaches.

Free Boundary Problems Involving Electrically
Charged Viscous Fluids.
Free boundary problems involving electrically charged fluids and
the presence of surface tension forces are attracting the attention of broad communities of physicists and engineers. The main
reason is the possibility of controlling the behaviour of fluids at
micro and nanometer length scales by means of electric and
magnetic fields with all the potential applications that should
provide. Mathematically, these problems involve solving Stokes
equations in the fluid domain sub ject to boundary conditions
that balance viscous stresses with surface tension forces and
electrostatic repulsion of charges. If the fluid is partly in contact
with a solid, then noslip boundary conditions are imposed at
the solidfluid interface. The presence of repulsive, and hence
destabilizing, forces at the interface that oppose the stabilizing
surface tension forces gives rise to various interesting phenomena. These include: 1) the existence of nonspherical equilibrium
configurations for levitating drops that can be characterized as
symmetrybreaking bifurcations of spherical configurations, 2)
the existence of instabilities of the interface leading to the formation of finitetime geometrical singularities in the form of cones
and emission of jets, 3) the phenomenon of electrowetting, consisting in the control of the wetting properties of fluids by means
of electric fields. Mathematically, problem 1) is addressed via
CrandallRabinowitz's theorems suitably adapted to the study
of free boundary problems, problem 2) is studied via boundary
integral formulations and local analysis near the singularities
and problem 3) is essentially variational. We will review recent
contributions as well as some open problems.

Optimal Design of Thin Plates
We consider an optimal design problem, which consists in finding the optimal distribution of a prescribed amount of platelike
material in a certain design region, in order to minimize the compliance under a given system of forces. We identify admissible
mass distributions to positive measures with prescribed integral
mean, thus allowing both diffused and concentrated solutions.
By this way, we immediately get the existence of an optimal
design, and the minimal compliance can be recovered simply by
maximizing a linear form under an Hessian constraint. I will
discuss how this model can be derived from 3Dlinear elasticity,
and I will give necessary and sufficient optimality conditions,
which can be used in order to compute the minimal value of the
compliance and to determine analytically some optimal plates.
The results are contained in some recent joint works with G.
Bouchitte.

On the Convexity of Some Free Boundaries
This talk deals with the free boundary arising in some varia
tional problems, with or without obstacle, including the well
known dam problem. The free boundary is the set { x : u(x) =
0 }, where u minimizes a convenient convex functional in some
convex space of admissible functions. In a work in progress
with B. Kawohl, u is compared via maximum principle with its
quasiconcave envelope u^{∗} (the smallest quasiconcave function
above u). It turns out that u = u^{∗} and therefore every level
set of u is convex. A similar approach was formerly used by
Colesanti and Salani (2003) and by Cuoghi and Salani (2006) in
the case when the set {u = 0} is prescribed.

The Cylinder Is Not the Optimal Shape for a Pipe
In this talk, we are interested in the optimal shape of a pipe
(inlet and outlet are fixed and the volume is prescribed). We
consider an incompressible fluid, subject to the NavierStokes
equations with classical boundary conditions on the boundary
of the cylinder (velocity profile given on the inlet, no slip con
dition on the lateral boundary and outletpressure condition on
the outlet). We are interested in the following question: is the
cylinder the optimal shape for the criterion "energy dissipated
by the fluid"? We prove that it is not the case. For that pur
pose, we explicit the first order optimality condition, thanks to
adjoint state and we prove that it is impossible that the adjoint
state be a solution of this overdetermined system (joint work
with Yannick Privat).

From Splish to Splash: Aspects of Water Impact Problems
I shall describe asymptotic approaches to a range of 'violent impact' problems involving collisions between liquid and solid or liquid bodies. Applications include impact of a blunt body on water, early stages of droplet collison, and high Froude number shallowwater flows on a sloping base. Mathematical connections range from variational inequalities to delta shocks.

PDE free boundary problems in tumor models
We shall discuss the recent progress (joint work
with Avner Friedman) on the PDE tumor models, the linear stability
of the tumor, the nonlinear stability of the tumor, and the comparison
of the stability between different models. These comparisons
have implications on the physical problems.

The Supercooled Stefan Problem in One Dimension
We investigate the supercooled Stefan problem
where V
_{x,t} denotes the outward velocity of the free boundary ∂{ρ > 0} with
respect to the positive phase. We will discuss corresponding discrete model
using interacting particles, corresponding conservation laws and properties of
the weak solution depending on the size of the initial data. This is joint work
with Lincoln Chayes.

Multiphase modelling of biological tissue growth.
Tissue engineering raises some novel modelling challenges and
some continuum models developed to describe aspects of the
growth process will be described. Their relationship to freeboundaryproblem and nonlinearPDE formulations that arise
in other contexts will be mentioned.

Singular Solutions of Mixed Boundary Value Problems of Water Impact
Initial stage of plate impact onto a liquid free surface is considered in
twodimensional and axisymmetric configurations. The ratio of the plate
displacement to the plate dimension plays the role of a small parameter.
Asymptotic analysis is performed within the ideal and incompressible liquid
model. Method of matched asymptotic expansions is used to derive the
second order outer solution, which is valid outside small vicinities of the
plate edge, and the equations which govern the flow close to the plate edge.
It is shown that the initial flow close to the plate edge is twodimensional
and selfsimilar in the leading order. This flow is governed by nonlinear
boundaryvalue problem with unknown shape of the free surface. The inner
velocity potential was calculated numerically by boundaryelement method.
The farfield asymptotics of the inner solution is analytically derived. This
asymptotics reveals an eigen solution which gives rise to important singular
contribution to the secondorder outer velocity potential. In addition to
this singular eigen solution the secondorder velocity potential is singular
on its own at the plate edge. The singular solution of the outer secondorder
problem is obtained with the help of theory of analytic functions in twodimensional case and a combination of special functions in the axisymmetric
case. The secondorder outer solution is matched with the leading order
inner solution. Uniformly valid distribution of the hydrodynamic pressure
over the plate is obtained. Initial asymptotic behaviour of the total
hydrodynamic force acting on the entering plate is analytically evaluated
and compared with the results of numerical calculations by nonlinear
potential solver. It is shown that the eigen solution of the outer problem
gives the most important contribution to the total force.
The eigen solutions of the plate impact problem are used to derive the
secondorder velocity potential and the pressure distribution in the problem
of wave impact onto an infinite rigid plate. In the latter problem the wetted
area of the plate grows in time and the leading order inner solution is known
in analytical form.

Crust Formation in Bread Baking
A model for the baking of bread, and in particular the formation
of a crust during baking, is presented. The crust is the outermost part of a bread loaf where the final density is significantly
higher than in the crumb, the main part of the loaf. Our model
is based on a collapse mechanism, whereby raised pressures in
the interior of the loaf, due to thermal expansion and water
evaporation, squash bubbles in the outer part of a bread loaf
at the same time as the bread sets and fractures. The latter
process allows vapour to escape from bubbles which can then
shrink. The bubble collapse occurs at a free boundary which
moves into the bread loaf. A second free boundary models the
boiling off of water.

American Options and Parabolic IntegroDifferential Operators.
The price of an American option is generally computed by solving an optimal stopping problem. In this talk, based on joint work with Mohammed Mikou, we will first discuss the connection between optimal stopping
of Lévy processes and parabolic integrodifferential inequalities. We will then
focus on some qualitative properties of the American put price in exponential Lévy models. In particular, we will study the associated free boundary, which is related to the optimal exercise strategy. We will also investigate the
socalled smooth fit property in this setting.

Viscosity Method in Homogenization
In this talk we are going to consider viscosity method in the
homogenizations
of the highly oscillating elliptic or parabolic obstacle problems
and of some nonlinear problems in perforated domain.
This viscosity method can be applicable for the fully nonlinear equations.
For the simplicity, we may consider obstacles that are consisted of
cylindrical columns distributed periodically. If the decay rate of the
capacity of columns is too high or too small, the limit of u_{ε}
ends up with trivial solutions.
The critical decay rates of having nontrivial solution are obtained
with construction of barriers. And discrete gradient estimate and almost
flatness in each cell
will give us a concept of convergence.
We also show the limit of u_{ε} satisfies a homogenized equation with a
term showing the effect of the
highly oscillating obstacles in viscosity sense.
We will also discuss the other issues in different nonlinear problems.

Modelling Frying Processes
A model for the deepfrying of an undeformable potatoslice is
discussed.
The derivation of the model from the basic principles of mass and
enthalpy conservation is presented. Attention is focused on the early
stage of vaporization, for which an explicit solution is provided,
thus bypassing the computational difficulties connected with the
appearance of an interface. We present numerical simulations that
show a satisfactory agreement with the available experimental data.

Regularity of a Free Boundary in Parabolic Problem Without Sign Restriction
We consider a parabolic obstacletype problem without sign restriction on the solution, for which we obtain the exact representation of the global solutions (i.e., solutions in the entire
halfspace {(x, t) ∈ R^{n+1}: x_{1} > 0}) and study the local properties of the free boundary near a fixed one. We also prove the
smoothness of the free boundary under a homogeneous Dirichlet
condition on the given boundary. This is a joint work with D.
Apushkinskaya and N.N. Uraltseva.

New Free Boundary Problem in Liquid Filtration
Arising Via Homogenization
In the present talk we consider new free boundary problems describing a joint
filtration of two immiscible incompressible viscous fluids with different viscosities
and densities. On the microlevel the mathematical model consists of Stokes equation for the liquid velocity in the pore space, Lame equation for the displacements
of the elastic solid matrix, continuity conditions on the joint boundary "liquidsolid" and transport equations for the unknown density and viscosity of the liquid.
This problem is very hard to tackle due to its nonlinearity and the fact that its
main differential equations involve nonsmooth oscillatory coefficients, both big
and small, under the differentiation operators. To simplify the model we suggest
a homogenization procedure when the dimensionless size ε of pores tends to zero,
while the porous body is geometrically periodic. For the single fluid the structure
of homogenized equations depend on the limits:
where μ is the viscosity of fluid, λ is the elastic Lamé constant,
L is a characteristic
size of the domain in consideration, τ is a characteristic time of the process, ρ
_{0} is
the mean density of water, and
g is the value of acceleration of gravity. There are
two scenarios to get new homogenized free boundary problems. The first scenario
proposes a limiting procedure as ε ↓ 0 for free boundary problems on the micro
level. As a result we obtain a new model together with its solvability. According
to the second scenario we first fulfill the passage to the limit as ε ↓ 0 for each
domain, occupied by single liquid, and then add the transport equations for the
density and viscosity of the liquid to already obtained homogenized equations,
describing the motion of the medium. Both ways give the same mathematical
models, but for the second scenario the correctness of the obtaned model remains
open. Namely, if
1) μ
_{0} = 0, 0 < μ
_{1} < ∞ and λ
_{0} = ∞ we arrive at the wellknown Muskat
problem, which is still unsolved. If
2) μ
_{0} = 0, μ
_{1} < ∞ and 0 < λ
_{0} < ∞, then we arrive at new TerzaghiBiotMuskat problem, which consist of the TerzaghiBiot system for the filtration of a
viscous liquid in the elastic solid skeleton coupled with the transport equations for
the density and viscosity of the liquid. Finally, if
3) 0 < μ
_{0} , λ
_{0} < ∞ then the limiting regime is described by the system of viscoelasticity for the displacements of the medium coupled with the corresponding
transport equations for the density and viscosity of the liquid. For this last problem
we prove the existence and uniqueness of the generalized solution.

Session: Homogenization
Random Homogenization of Fractional Obstacle Problems
We study the homogenization of fractional obstacle problems in a
perforated domain, when the holes are periodically distributed and have
random shape and size. The main assumption is that the "size" of the
holes is stationary ergodic. Fractional obstacle problems arise, for
instance, when modeling molecules transport across semipermeable membranes.
Session: Combustion
Fronts Propagation in Inhomogeneous Media
I'll review various results concerning combustion fronts propagation in
periodic and non periodic media.

On The Fully Nonlinear Signorini Problem
We will discuss recent results on free boundary problems for
fully nonlinear equations which are motivated by applications
in elasticity, such as the fully nonlinear Signorini problem. We
will concentrate on the tools which will lead us to regularity of
solutions for those free boundary problems, in which the free
interface has an active role in the overall process. This is a joint
work with L. Silvestre.

Reviving Wagner's theory for highvelocity solidliquid impact.

Continuous Interface With Disorder: Even Strong
Pinning Is Too Weak in Two Dimensions
I present recent results about a statistical mechanics model of
continous height effective interfaces in the presence of a deltapinning.

Numerical study of an optimal partition problem.
We introduce a new numerical approach in order to approximate the partition which minimizes the sum of its fundamental
modes.

The British Option
We present a new put/call option where the buyer may exercise
at any time prior to maturity whereupon his payoff is the 'best
prediction' of the European payoff under the hypothesis that
the true drift of the stock price equals a contract drift. Inherent in this is the protection feature which is key to the British
option. Should the option holder believe the true drift of the
stock price to be unfavorable (based upon the observed price
movements), he can substitute the true drift with the contract
drift and minimize his losses. With the contract drift properly
selected the British put option becomes a more 'buyer friendly'
alternative to the American put: when stock price movements
are favorable, the buyer may exercise rationally to very comparable gains; when price movements are unfavorable he is afforded the unique protection described above. Moreover, the
British put option is always cheaper than the American put.
In the final part we present a brief review of optimal prediction
problems which preceded the development of the British option.
This is a joint work with F. Samee (Manchester).

A NonStandard Free Boundary Problem in Pasta Cooking
The talk will describe a work in progress in which a free boundary problem for the diffusion equation is studied modelling the cooking of pasta. Water infiltration and gelatinization seem to be the most relevant processes occurring during the cooking; their interplay originates a problem which is nonstandard and presents some interest in itself.

Monotonicity Formulas and the Singular Set in the
Thin Obstacle Problem.
We construct two new oneparameter families of monotonicity
formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type
formulas geared for points of any given homogeneity and the
second one is a family of Monneau type formulas suited for the
study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given
homogeneity. This allows to prove a structural theorem for the
singular set.
Our approach works both for zero and smooth nonzero lower
dimensional obstacles. The study in the latter case is based on a
generalization of Almgren's frequency formula, first established
by Caffarelli, Salsa, and Silvestre.
This is a joint work with Nicola Garofalo.

Traveling Waves in a HeleShaw Type Moving
Boundary Problem
We discuss a 2D movingboundary problem for the Laplacian
with Robin boundary conditions in an exterior domain. It arises
as model for HeleShaw flow of a bubble with kinetic undercool
ing regularization and is also discussed in the context of models
for electrical streamer discharges.
The corresponding evolution equation is given by a degener
ate, nonlinear transport problem with nonlocal lowerorder de
pendence. We identify the local structure of the set of traveling
wave solutions in the vicinity of trivial (circular) ones. We find
that there is a unique nontrivial traveling wave for each velocity
near the trivial one. Therefore, the trivial solutions are unstable
in a comoving frame.
The degeneracy of our problem is reflected in a loss of regu
larity in the estimates for the linearization. Moreover, there is
an upper bound for the regularity of its solutions.
(Joint work with M. Günther, Universität Leipzig)

Remarks on a Class of Two Phase Free Boundary Problems
We present some properties of the solution to a class of free boundary
problems with two phases. Under a natural nondegeneracy assumption on the
interface, for which a sufficient conditon is given, we prove a continuous
dependence result for the characteristic functions of each phase and we
establish sharp estimates on the variation of its Lebesgues measure with
respect to the L^{1}variation of the data, in a rather general framework.

Free Boundary Problems for the Fractional Laplacian
We discuss here local properties  optimal regularity, nondegeneracy, smoothness
 of a free boundary problem involving the fractional Laplacian, generalising the
classical phase transition problem for the standard Laplacian with gradient jump.
Our equations are relevant models for boundary reactions, but also to reaction
diffusion processes involving nonGaussian diffusion.
The nonlocality of the fractional laplacian renders the problem nontrivial, and the
key tool is the CaffarelliSilvestre extension formula, which transforms the model
into a codimension 2 free boundary problem.
Joint work with L. Caffarelli and Y. Sire.

Species Persistence: the Optimal Habitat Shape
n a binary environment made of habitat and nonhabitat regions, we study species persistence
through a reactiondiffusion model. Species survival both depends on habitat abundance and shape. For a fixed habitat abundance, we describe the shapes that maximize the chances of survival, in bounded and infiniteperiodic environments.

Geometric Methods for the Convergence of Diffuse
Interface Models
Phase field models are a common approach to phase separation processes. They are often given as gradient flows of a diffuse surface energy that goes back to van der Waals and CahnHilliard. Often these models correspond formally to a sharp
interface limit. To justify the passage to the limit techniques
from geometric measure theory have proved useful. We report
on progress and limitations of this approach.

Traveling Wave Solutions for a NonConvex Fpu
Lattice Model Connecting Different Phases
The existence of travelling waves in atomistic models for marten
sitic phase transitions is studied. The elastic energy is assumed
to be piecewise quadratic, with two wells representing two stable
phases. The focus is on subsonic waves with strains exploring
both wells of the energy. We prove rigorously the existence of
travelling waves in a onedimensional chain of atoms. The wave
is 'heteroclinic' in the sense that asymptotically the strains are
contained in different wells of the energy.

Advanced Technique of Complex Variable for Solving Nonlinear Free Boundary Flows
Historically, a progress in solving problems of twodimensional free boundary potential flows is based on a development of the theory of complex variable. Since any analytical function meets the requirements of a fluid incompressibility and zero vorticity, the problem is to find such analytical function which satisfies to given boundary conditions.
The present talk will be focused on determination of a complex function from its modulus and argument or its real part and argument, which are given on the boundary of a simply connected domain. In combination with Chaplygin's singular point method it makes possible to determine expressions for a complex velocity and for a derivative of the complex potential of an arbitrary unsteady free boundary flow.
These expressions contain in an explicit form the modulus and argument of the velocity defined as functions of a parameter variable and time. The dynamic and kinematic boundary conditions lead to a system of integral and integrodifferential equations for determination of these unknown functions.
The proposed method recently been applied to the solution of selfsimilar water entry problems and timedependent problems of HeleShaw flows with and without surface tension.

Complex Singularities in Interfacial Fluid Flow and
3D Incompressible Euler Equations.
One of the most interesting phenomena in free surface flows is
singularity formation on the interface. Examples include topological singularities, such as pinchoff of a liquid thread, or the
formation of a curvature singularity on an evolving vortex sheet.
In the first part of this talk, we describe how singularity formation can be understood by analytically extending the variable
which parameterizes the interface and analyzing the singularity
motion in the complex plane. As an example, we discuss the
recent construction of singular solutions for twophase flow in
porous medium or HeleShaw cell. In the second part, we describe a new approach for the construction of complex singular
solutions to the 3D Euler equations that is motivated by the
results for interfaces.

Regularity Results for the Obstacle Problem for the Fractional
Laplacianusing thin Obstacle Problems.
We obtain the optimal regularity of the solutions and the regularity of the
free boundary of the obstacle problem for the fractional Laplacian. We
rewrite the original nonlocal problem as an equivalent local problem in one
dimension more. The obstacle problem for the fractional Laplacian turns out
to be equivalent to an appropriate "thin" obstacle problem. In this way we
can study a problem involving the fractional Laplacian using standard PDE
(local) tools. This is a joint work with Luis Caffarelli and Sandro Salsa.

The Energy + Dissipation Functional.
The evolution of dissipative systems is governed by the interplay between energy and dissipation. Roughly speaking, evolution is driven by energy whereas dissipation sets the reference
metric landscape (in other words, energy decreases along dissipation geodesics). Although energy and dissipation play such
substantially different roles in the evolution, they get summed
together and minimized in the standard implicit Euler timediscrete scheme.
The purpose of this talk is to report on a recent progress
in the variational treatment of the dissipative evolution which
basically consists in a timecontinuous version of this fact. In
particular, I will focus on a class of globalintime functionals
recently proposed by Mielke & Ortiz where the (weighed) sum
energy+dissipation comes directly into play. The interest in this
perspective is that of possibly applying the tools of the Calculus
of Variations (e.g., Γconvergence, relaxation, approximation)
to evolution. I will review some flow and the rate independent
case.
This is a joint project with A. Mielke (Berlin).

Effect of Regularization on Stability of Simple Steadily
Propagating Shapes in Streamer Fronts and Other Related Problems.
An important problem in pattern formation is the stability of
propagating shapes. In the context of streamers, modeled as a sharp front
where equations similar to that of quasisteady crystal growth with
kinetic regularization arise, previous work by Brau et al show steadily
propagating circular shapes are linearly stable at a particular value of
kinetic regularization. Recent work with collaborators will be presented
on the spectral analysis of the linear stability problem for arbitrary
value of the regularization parameter. It is shown that all eigenvalues
are negative. We compare and contrast these results with earlier work on
the mathematically related viscous fingering problem with surface tension
for circular and more general shapes. In the viscous fingering case, we
also present some rigorous mathematical results on the nonlinear stability
problem.

A Two Parameter Family of Expanding Wave
Solutions of the Einstein Equations that Includes the
Standard Model of Cosmology.
In this talk I discuss recent joint work with Joel Smoller in which
we derive an exact two parameter family of expanding wave solutions of the Einstein equations that includes the critical (flat
space k = 0) Friedmann universe in the standard model of cosmology. All of the spacetime metrics associated with this family
apply when the equation of state is given by p = c^{2} /3 ρ, correct for early Big Bang physics, after inflation. By expanding
solutions near the standard model, about the center, to leading order in the Hubble length, we find a oneparameter family
of expanding spacetimes that represent a perturbation of the
standard model. We then show that there exists a coordinate
system in which the perturbed spacetimes agree with the standard model in each spacelike timeslice, (a flat metric with a
time dependent scale factor), but there are small corrections
to the Hubble constant, the particle velocities, and there is a
small spacetime cross term. Since exact noninteracting expansion waves represent possible timeasymptotic wave patterns for
conservation laws, we wonder whether it is possible that these
corrections to the standard model might account for the anomalous acceleration of the galaxies, without the introduction of the
cosmological constant. (Articles and commentaries can be found
on author's website: http://www.math.ucdavis.edu/ temple/articles/) .

Characteristic Discontinuities in Magnetohydrodynamics
In ideal compressible magnetohydrodynamics (MHD) there are three types of
characteristic
discontinuities: currentvortex sheets, contact discontinuities, and Alfven
discontinuities.
We survey recent results in the study of existence and stability of such
characteristic free
boundaries for the MHD equations. The main attention is devoted to
currentvortex sheets
but, for example, stability results for Alfven discontinuities are also
discussed.
The main result for currentvortex sheets is a localintime existence
theorem. Namely, we prove the
localintime existence of solutions with a surface of currentvortex sheet
of the MHD equations in three space
dimensions provided that a stability condition is satisfied at each point of
the initial discontinuity.
The fact that the KreissLopatinski condition is satisfied only in a weak
sense yields losses of derivatives
in a priori estimates. Therefore, we prove our existence theorem by a
suitable NashMosertype iteration scheme.

Some Results on a Class of Serrin Type Overdetermined Problems
We extend the classical symmetry result of Serrin to the case of Hessian equations presenting an alternative proof which does not rely explicitly on the maximum principle. The techniques used can be also adapted to obtain stability results with respect to perturbations of boundary data.

Crystal Dissolution and Precipitation in Porous Media: Formal Homogenization and Numerical Experiments
We investigate a twodimensional microscale model for crystal dissolution
and precipitation in a porous medium. The model contains a free boundary and
allows for changes in the pore volume. Using a levelset formulation of the free
boundary, we apply a formal homogenization procedure to obtain upscaled
equations. For general microscale geometries, the homogenized model that
we obtain falls in the class of distributed microstructure models. For circular
initial inclusions the distributed microstructure model reduces to system of
partial differential equations coupled with an ordinary differential equation.
In order to investigate how well the upscaled equations describe the behavior of
the microscale model, we perform numerical computations for a test problem.

Some New Three Dimensional Free Surface Flows
Over the last 200 years many analytical and numerical results have been obtained
for two dimensional free surface flows. A large part of this success is related to the fact
that two dimensional free surface flows can be formulated in terms of analytic functions.
Relatively few solutions have been calculated for three dimensional free surface flows.
In this talk we present new three dimensional free surface flow solutions. These include
gravity capillary solitary waves and flows generated by moving disturbances.

Porous Medium Flow with Nonlocal Effects
We study a model for flow in porous media including nonlocal
diffusion effects. It is based on Darcy'sm law and
the pressure is related to the density by an inverse
fractional Laplacian operator.
We prove existence of solutions that propagate with finite speed.
The model has the very interesting property that mass preserving
selfsimilar solutions
can be found by solving an elliptic
obstacle problem with fractional Laplacian for the pair pressuredensity.
We use entropy methods to show that the asymptotic behaviour is described
after renormalization by these solutions which play the role of the
Barenblatt profiles of the standard porous medium model.

A Nonlinear Frequency Formula and the Singular set of a Free Boundary Problem
The degenerate singular set Σ_{z} of the Bernoulli free boundary problem is known to be a nullset with respect to the ndimensional Lebesgue measure.
Here we show that Σ_{z} can be decomposed into a Reifenberg flat part and a lower dimensional nullset.
The key is a formula for a nonlinear mean frequency. We discuss generalizations to other equations.

Regularity of the Free Boundary for the Limit of an Inhomogenous Singular
Perturbation Problem
In this lecture we present several results on the regularity of the free boundary for the limit
u = lim u
^{εj} ; with ε
_{j} →0 and u
^{εj} solutions to
where
.
We prove that, under certain assumptions, u is a weak solution to the free boundary problem
where M = ∫β(s) ds, with β a Lipschitz continuous function, β(s) > 0 for s ∈ (0,1), β(s) = 0
for s ∉ (0,1).
The term f
_{εj} may be a true source term or may come from lower order terms that are already
known to be bounded uniformly with respect to ε
_{j} . It may also come from a nonlocal diffusion term.
The proofs make use of the regularity results for two phase "viscosity solutions" of the inhomogeneous stationary problem by Caffarelli, Jerison and Kenig.
Also, we prove a
local monotonicity formula for the inhomogenous case inspired by a global
one by G. S. Weiss for the homogeneous case (this is, f
_{εj} ≡ 0). This formula is a main tool in some of our results.

ShockFree Periodic Solutions for the
Euler Equations
In this ongoing collaboration with Blake Temple, we attempt to prove
the existence of periodic solutions to the Euler equations of gas
dynamics. Such solutions have long been thought not to exist due to
shock formation, and this is confirmed by the celebrated GlimmLax
decay theory for 2x2 systems. However, in the full 3x3 system,
multiple interaction effects can combine to slow down and prevent
shock formation. In this talk I shall describe the physical mechanism
supporting periodicity, analyze combinatorics of simple wave
interactions, and develop periodic solutions to a "linearized"
problem. These linearized solutions have a beautiful structure and
exhibit several surprising and fascinating phenomena. I shall also
discuss our attempts to prove that these solutions perturb: this leads
us to problems of small divisors and KAM theory.

The Zero Level Set for Certain Weak Solutions with
Applications to the Bellman Equations.
We will investigate the zero level set of weak solutions to
div(A(u) ∇u) = 0, (1)
where A(t) = I when t ≥ 0 and A(t) = B when t < 0. Our interest in this problem comes from its relation to the Bellman equations, max(Δv, div(B ∇v)) = 0.
In particular, there is an equivalency between the zero level set of solutions
to (1) and the set where v changes phase, from solving Δv = 0 to solving
div(B ∇v) = 0.
We will show that the zero level set has σfinite (n1)dimensional Hausdorff measure and that the zero level set is C^{1,α} almost everywhere with respect
to the measure Δu^{+}.
The proof relies on methods developed by L.A. Caffarelli for two phase free
boundary problems and some results by J.M. Marstrand in geometric measure
theory.
This is a joint work with H. Mikayeyan.

An Approximation Technique for the LaplaceYoung
Capillary Equations in a Circular Cusp Region
Solder plating of a semiconductor pin is one of the industrial
applications of capillary free boundary problems. The capillary
filling in a circular cusp region has been explored employing the
LaplaceYoung equations. In this problem, the tangent cylindrical coordinate system has shown a significant advantage over
the previously used curvilinear coordinate system. In order
to approximate the solution to the LaplaceYoung equations,
modified LaplaceYoung Equations are constructed. It can be
shown that the use of the tangent cylindrical coordinate system
make it possible to solve the modified LaplaceYoung equations
exactly. Also the exact solution to the modified LaplaceYoung
equations is shown to be a seventh order accurate asymptotic
solution to the original LaplaceYoung equations. This talk will
address how a slight modification to the PDE problem and a
proper choice of the coordinate system dramatically simplified
a problem and unveiled an accurate approximation.

Nonlinear PDE methods in terms of stochastics
We look at methods used in nonlinear PDE theory and their applica
tions to finance. The blowup technique for instance can be used to obtain
boundary regularity of nonlinear free boundary problems.
The PDE problem has a corresponding formulation in terms of stochas
tic, but we do not find the corresponding method for local analysis of the
stochastic problem. Our aim is translate nonlinear PDE methods into
stochastic language. We show how this is done for the linear optimal
stopping problem of pricing American options.

Numerical algorithms for the spatial segregation of competitive
systems.
A system of m differential equations that appears in population modeling,
is considered. At first we study the asymptotic behavior of the positive
solutions as the competition rate tends to infinity, then we present
various numerical methods.

Motion of Hypersurfaces by the Harmonic Mean
Curvature Flow
We will analyze the evolution of weakly convex surfaces in R^{3}
with flat sides by the harmonic mean curvature flow. We establish short time existence as well as the optimal regularity of
the surface and show that the boundaries of the flat sides evolve
by the curve shortening flow. It follows from our results that
a weakly convex surface with flat sides of class C^{k,γ}, for some
integer k ≥ 1 and 0 < γ ≤ 1, remains in the same class.

Free Boundary Behaviour of Finitely Liquid Markets
A number of models have been proposed with the aim of incorporating finite liquidity and price impact into option pricing
theory. For the most part these result in highly nonlinear partial differential equations, on account of the inherent feedback
mechanisms involved. We employ novel analytical and numerical techniques, including local similarity analysis, to evaluate
the option price behaviour and in particular the behaviour of
the early exercise boundary of American style options. Of particular interest is the period close to expiry of the early exercise
options, generally the most intricate time for the freeboundary
behaviour. The inclusion of finite liquidity reveals subtle differences from the infinitely liquid BlackScholes case.

Discrete time, finite state space mean field games
We study a mean field model for discrete time, finite number of states,
dynamic games. The mean field approach models situations that involve
a very large number of agents which move according to certain optimality
criteria.
We discuss the set up of the problem, as well as its connection with
discretetime finite number of states optimal control problems. In contrast
with optimal control, where usually only the terminal cost V^{N} is necessary to
solve the problem, in meanfield games both the initial distribution of agents
π^{0} and the terminal cost V^{N} are necessary to determine the solutions, that
is, the distribution of players π^{n} and value function V^{n} , for 0 ≤ n ≤ N .
Because both initial and terminal data needs to be specified, we call this
problem the initialterminal value problem. We establish existence, under
quite general conditions, as well as some uniqueness results, both for the
stationary and for the initialterminal value problems. Finally we prove the
exponential convergence to a stationary solution of (π^{0} , V^{0}), as N → ∞, for
the initialterminal value problem with data π^{N} and V^{N} .
This is a joint work with R. R. Sousa and J. Mohr.

Multiplicity Results to the Existence of Three Solutions for a Class of Neumann Elliptic Systems
In this work, we are interested in multiplicity results for the following
Neumann elliptic systems
where Δ
_{pi} u
_{i} =div(∇u
_{i}
^{pi2} ∇u
_{i} ) is the p
_{i}Laplacian operator, p
_{i} > N for
1 ≤ i ≤ n, Ω ⊂ R
^{N}(N ≥ 1) is nonempty bounded open set with a boundary
∂Ω of class C
^{1} , p
_{i} ≥ 2, a
_{i} ∈ L
^{∞}(Ω) with ess inf
_{Ω} a
_{i} > 0 for 1 ≤ i ≤n,
λ, μ > 0, F : Ω × R
^{n} → R is a function such that F (., t
_{1}, ..., t
_{n}) is continuous
in Ω for all (t
_{1} , ..., t
_{n} ) ∈ R
^{n} and F(x, ., ..., .) is C
^{1} in R
^{n} for almost every
x ∈ Ω, G : Ω × R
^{n} → R is a function such that G(., t
_{1} , ..., t
_{n}) is measurable in
Ω for all (t
_{1} , ..., t
_{n} ) ∈ R
^{n} and G(x, ., ..., .) is C
^{1} in R
^{n} for almost every x ∈ Ω,
and F
_{ui} and G
_{ui} denotes the partial derivative of F and G with respect to
u
_{i} , respectively, and ν is the outward unit normal to ∂Ω.
Precisely, we deal with the existence of an nonempty open interval Λ ⊂ [0, +∞[ and a positive real number q with the following property: for every
λ ∈ Λ and every G : Ω × R
^{n} → R as above, there exists δ > 0 such
that, for each μ ∈ [0, δ], the problem (1) admits at least three solutions in
W
^{1,p1}(Ω) × W
^{1,p2}(Ω) × ... × W
^{1,pn}(Ω) whose norms are less than q.
Very recently, B. Ricceri in [2] revisited the three critical points theorem
of [1], and our result is fully based on it.
Keywords Three solutions; Critical point; (p
_{1} , ..., p
_{n} )Laplacian; Multiplicity results; Neumann problem.
AMS subject classification: 35J65; 34A15.
[1] B. Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75
(2000) 220226.
[2] B. Ricceri, On a three critical points theorem revisited, Nonlinear AnaL.
To appear.

On Collisions Between Rigid Bodies Inside a Viscous
Incompressible Fluid
In order to describe interactions between rigid bodies in a viscous incompressible uid, a common approach consists in writing
incompressible Navier Stokes equations in the fluid domain, with
no slip boundary conditions, and applying classical mechanics
relations to describe the dynamics of the solids inside the fluid.
At the end of the 90s, several studies prove existence of solutions to such problems up to collision between solids (see B.
Desjardins and M.J. Esteban, CPDE, (25)2000, pp. 13991413,
for example). In the following years, several attempts were made
to determine whether these models allow contact or not. First,
J.L. Vazquez and E. Zuazua (M3AS, (5), 2006, pp. 637678)
prove a no collision result in a simplified 1D problem. Then,
V.N. Starovoitov (FBP, Trento, 2002) obtains a criterion for
the lack of collision involving sobolev norms of the uid velocityfield gradient. However, this criterion does not apply a priori
to solutions to the NavierStokes equations. In my talk, I shall
explain how the strategy by V.N. Starovoitov can be improved
to obtain a rigorous no collision result and which part of the
model may be loosen to allow collisions.

Anomalous LargeTime Behaviour of the PLaplacian
Flow in an Exterior Domain in Low Dimension
We study the large time behavior of weak nonnegative solutions
of the pLaplace equation (p > 2) posed in an exterior domain
in space dimension N < p with homogeneous Dirichlet bound
ary conditions, together with the behavior of their free bound
aries. The description is done in terms of matched asymptotics:
the outer asymptotic profile is a dipolelike selfsimilar solution
with a singularity at x = 0 and anomalous similarity exponents,
while the inner asymptotic behavior is given by a separate
variable profile. As intermediate interesting results, we prove
a new strong maximum principle for the pLaplacian equations
near degeneracy points. Moreover, we use a fine geo metric
technique to study the evolution of the free boundary, passing
through delicate comparison arguments.
Joint work with Prof. Juan Luis Vazquez.

Preisach Model in Hydrology
We present results about a class of PDEs whose model equa
tion is represented by the PhilipRichards equation with soil
moisture hysteresis term. We assume that the porous media
hysteresis is represented by the Preisach hysteresis operator.
We introduce a weak formulation of our problem in Sobolev
spaces. An existence result is proved by a method based on an
ap proximation of implicit time discretization scheme, apriori
estimates and passage to the limit in the convexity domain of
the Preisach operator.

Regularity of the Boundary of an Optimal Shape With
Convex Constraints
We deal with the following shape optimization problem:
where λ
_{2} denotes the second eigenvalue of the Laplacian with
Dirichlet boundary conditions (in dimension 2).
A. Henrot and E. Oudet studied some properties of optimal
shapes Ω
^{*} in [HO]. They prove in particular that the stadium is
not optimal. They obtained other geometric properties of Ω
^{*},
under some extra assumptions about the regularity of this one
(for example: Ω
^{*} is C
^{1,1} ).
Precisely, we study here the regularity of the boundary of Ω
^{*} .
We will see that Ω
^{*} is at most C
^{1,1/2}, and not C
^{1,1} as expected
in the work of A. Henrot and E. Oudet. We also explain that,
with a few ajustments, the results of [HO] are still available.
Our method can in fact be applied to "partially overdetermined
problems", that is to say an equation like:
where Γ ⊃ ∂Ω and u ∈ C
^{1}(Ω). We will see the link with the
previous question, and we will show that our approach carries
over some other problems.
[HO] Henrot A.  Oudet E.  Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions, Arch. for rat. mech. and analysis 2003, vol. 169,
1, pp. 7387
[FG] Fragal I.  Gazzola F.  Lamboley J.  Pierre M.
C ounterexamples to symmetry for partially overdetermined
elliptic problems , Preprint, 2008
[L] J. Lamboley, About Hölder regularity of the optimal convex planar shape for λ
^{2} , Preprint, 2008

A 2Phase Free Boundary Problem for a Nonlinear
DiffusionConvection Equation
The RosenFokasYorstos equation is a well known model for
the two phase flow of immiscible fluids in a porous medium.
We consider a 2phase free boundary problem for such equation
and via a contraction mapping technique we show existence and
unicity of the solution for short time. An explicit solution is
also discussed

On the twophase obstacle problem with Hölder
continuous coefficients
We study the regularity of twophase obstacle problem, i.e.
Δu = λ1 χ{u>0}  λ2 χ{u<0} in B_{1}
with coecients λ
_{i} that merely Hölder continuous. When the coecients are
assumed to be Lipschitz one can apply the monotonicity formula to prove C
^{1,1}regularity of the solution and also the C
^{1}regularity of the free boundary near
so called branching points. However, there is no available monotonicity formula
in this case. Instead we use a specific scaling argument to obtain C
^{1,1}estimates
and then the stability in coecients to prove the C
^{1}regularity of the free boundary near so called branching points.
This is an ongoing project with Anders Edquist and Henrik Shahgholian,
both at KTH.

A Singular Perturbation Problem for a Quasilinear
Operator Satisfying the Natural Growth Condition of
Lieberman
We study the following problem. For any ε > 0, take u
_{ε} a weak
solution of
Here
with β ∈ Lip(R), β > 0 in (0,1) and β = 0 otherwise.
We assume that g satisfies the conditions introduced by G.
Lieberman in [Li1]. The conditions on the function g allow for a
different behavior at 0 and at ∞. Moreover the set of functions
that satisfy these conditions include inhomogeneous functions.
We are interested in the limiting problem, when ε→ 0.
As in previous work with
L = Δ or
L = Δ
_{p} we prove, under
appropriate assumptions, that any limiting function is a weak
solution to a free boundary problem (in the sense defined in
[MW1]). Moreover, for nondegenerate limits we prove that the
reduced free boundary is a C
^{1,α} surface. This result is new even
for Δ
_{p}. Finally, we give two examples in which we can apply the
regularity results. In both examples the nondegeneracy property
is satisfied by the limiting function.
Li1 G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations 16
(1991), no. 23, 311361.
MW1 S. Martínez and N. Wolanski, A minimum problem with
free boundary in Orlicz spaces, Adv. Math. (2008).

A New QuasiVariational Inequality for a Maxwell
System
We consider a stationary nonhomogeneous electromagnetic problem
motivated by the Bean's criticalstate model for superconductors.
We introduce the natural functional framework composed by divergentfree
$L^p$ functions having curl in L^{p} and null normaltrace and we prove
a PoincaréFriedrichs type inequality.
Imposing that the current density does not exceed some critical value,
that may depend on the magnetic field, the model becomes a
quasivariational inequality for which we are able to prove existence of
a solution.
(Joint work with José Francisco Rodrigues and Lisa Santos)

A Solution of Parabolic Free Boundary Problems by
Semilinear ReactionDiffusion Systems
A nonlinear degenerate parabolic problem of Stefan and porous
medium type is considered. The degeneracy of the diffusion
characterizes the presence of a free boundary between different phases and the solution exhibits a global lack of regularity
across the free boundary. Therefore, analysis and numerical
approximation of the problem are much harder than those of
mildly nonlinear problems and semilinear problems. To avoid
the nonlinearities of the diffusion with degeneracy of the diffusion, we propose a reactiondiffusion systems with solutions that
approximate those of the nonlinear problems. The reactiondiffusion system includes only a simple reaction and linear diffusion. Resolving semilinear problems is typically easier than
dealing with nonlinear diffusion problems such as the degenerate parabolic equations. Therefore, our ideas are expected to
reveal new and more effective approaches to the study of nonlinear diffusion problems. We also consider application of the
reactiondiffusion system to numerical methods for the nonlinear
degenerate parabolic problem. A discretetime scheme for the
nonlinear problem is proposed by means of the reactiondiffusion
system. We can obtain stability results and the optimal error estimates for the scheme. Numerical experiments are carried out
using a numerical algorithm based on the discretetime scheme.
The numerical results show the effectiveness and efficiency of
the algorithm.

Asymptotic Behavior of Solutions to the Dirichlet
Eigenvalue Problem in Perforated Domains of General
Structure
We study the asymptotic behavior of the solutions to the Dirichlet eigenvalue problem for the linear higher order elliptic operator in the sequence of perforated domains. No more complication arises from the complex structure of the domain in the
proof of the solvability of this problem but it is practically impossible to find its solution neither by analytical no by numerical
methods. In this case the homogenization methods are usually
used. Under several conditions on the perforated domain, one
can describe the leading term of the asymptotic expansion of
the solution of the initial problem in terms of solution of a new
problem considered in a simple domain.
The homogenization result for the linear higher order elliptic problem in domains of general structure was obtained in [1].
The asymptotic behavior of the Dirichlet problem for nonlinear
elliptic second order equations under sufficiently weak assumptions on perforated domain was obtained in [2]. The Dirichlet
problem for nonlinear higher order equations with a special ellipticity condition and without any geometrical assumptions on
the structure of perforation was studied in [3].
In this talk we consider a sequence of perforated domains of
a general structure. The assumptions on the perforations are
formulated in terms of Borelâ€™s measures. Following the lines of
[1, 2], we construct the limit problem for the leading term of the
asymptotic expansion of the solution to the Dirichlet eigenvalue
problem for linear higher order elliptic equations in perforated
domains.
Joint work with Šárka Nečasová.
1 Khruslov, E.Ya., The first boundary value problem in domains with a complicated boundary for higher order equa
tions, Mat. Sbornik 103(145), N 4(8), (1977), 614629.
2 Dal Maso, G., Skrypnik, I.V., Asymptotic behaviour of
nonlinear Dirichlet problem in perforated domains, Ann.
Mat. Pura Appl. IV, Ser.174 (1998), 1372.
3 Dal Maso, G., Skrypnik, I.V., Asymptotic behaviour of
nonlinear elliptic higher order equations in perforated domains, J. Anal. Math. 79, (1999), 63112.

Regularity of Free Boundaries in Twophase
Problems for the pLaplace Operator
Some time ago L. Caffarelli developed, in a sequence of three
papers, a celebrated theory for general twophase free boundary problems for the Laplace operator. In the first paper Lipschitz free boundaries were shown to be C^{1,γ}smooth for some
γ∈(0,1) and in the second paper it was shown that free boundaries which are well approximated by Lipschitz graphs are in fact
Lipschitz. Finally, in the third paper the existence part of the
theory was developed. Recently John Lewis and I have been able
to generalize the results in the first two papers to the pLaplace
operator when p=2, 1< p< ∞, and the purpose of this talk
is to briefly describe these results. Our generalizations beyond
the harmonic case, which corresponds to p=2, are nontrivial
due to the nonlinear and degenerate character of the pLaplace
operator. In particular, our results and arguments rely heavily
on a toolbox of techniques which John Lewis and I have developed in our studies of the boundary behaviour of pharmonic
functions in Lipschitz domain and in domains which are well
approximated by Lipschitz domains in the Hausdorff distance
sense. In particular, our results concerning general twophase
free boundary problems for the pLaplace operator represent a
'tour de force' of the techniques we have developed.

Behaviour of a Capillary Surface Near a Cusp
Consider two tangent vertical cylinders, of possibly different materials, immersed in a fluid bath. We investigate the resultant
equilibrium capillary surface near the cusp. When the contact
angles on the two cylinders are supplementary, then the solution is continuous up to the corner. When the contact angles
are not supplementary then the solution tends to plus or minus infinity and we determine the order of growth. Previously,
Markus Scholz (2003) studied the case of power law cusps with
nonsupplementary contact angles.

The Coupled Heat and Mass Transfer During the
Freezing and Sublimation Processes of HighWater
Content Food Materials
A coupled problem of heat and mass transfer during the solidification of highwater content materials (e.g. foods, moist soils,
acqueous solutions and vegetable or animal tissues) is analyzed.
When these kinds of materials are refrigerated two simultaneous
physical phenomena take place: liquid water solidifies (freeze),
and surface ice sublimates. Ice sublimation takes place at the
surface of highwater content systems when they are frozen uncovered or without an impervious and tight packaging material.
The rate of both phenomena (solidification and sublimation) is
determined both by material characteristics and cooling conditions. The sublimation process determines fundamental features of the final quality of foods and influences the structure
and usefulness of frozen tissues. Modeling these two simultaneous processes is very difficult owed to the coupling of the heat
and mass transfer balances as well as to the existence of two
phasechange fronts that advance with very different velocities.
The process when only a solidification front is present has been
extensively studied in literature. On the contrary, mathematical results in the case in which both freezing and ice sublimation appear are scarce and no analytical solution has been
found. Moreover no experimental data about temperature and
concentration profiles, and freezing and dehydration fronts are
available. On the other hand, theoretical models were proposed
to describe the heat and mass transfer in these processes but,
in most of the published works, only semiempiric or numerical
methods are used to investigate the properties of solutions.
As well as the description of the freezing process is concerned,
the material can be divided into three zones: unfrozen, frozen
and dehydrated. Freezing begins at the refrigerated surface, simultaneously, ice sublimation begins at the frozen surface and a
dehydration front penetrates the material. Normally dehydration velocity is much lower than that of the freezing front.
We consider a semiinfinite material with characteristics similar
to a very diluted gel (whose properties can be supposed equal to
those of pure water). The system has an initial uniform temperature equal to T_{if} and an uncovered flat surface which, at time
t = 0 , is exposed to the surrounding medium (with constant
temperature T_{s} lower than T_{if} , and heat and mass transfer coefficients h and K_{m}). We assume that T_{s} < T_{0}(t) < T_{if} where
T_{0}(t) is the unknown sublimation temperature. In order to calculate the evolution of temperature and water content in time,
we will consider a twophase free boundary problem for the temperatures T_{d} = T_{d}(x,t) (dehydrated region) and T_{f} = T_{f}(x,t)
(frozen region), the vapour concentration C = C(x,t) (dehydrated region), the free boundaries s_{d} = s_{d}(t) (sublimation
front) and s_{f} = s_{f}(t) (frozen front) and the temperature T_{0}(t)
at the sublimation front x = s_{d}(t) .
We developed a quasianalytical model under few simplifying
assumptions, thus obtaining a set of coupled ordinary differential equations which enables to predict, easily, the influence of
material characteristics and freezing conditions on the evolution
of both frozen and sublimation fronts. In the same way we are
able to find the temperature in the frozen and dehydrated zones,
the temperature at the sublimation front and the vapour concentration in the dehydrated layer. Such results were validated
against the analytical solution describing the freezing process in
a semiinfinite material when the sublimation does not occur.
Our quasianalytical model has been extensively used to perform a parametric study of the problem and, in the future, it
will be used to optimize the freezing conditions of certain types
of materials.
This is a joint paper with Rodolfo H. Mascheroni, Mariela C.
Olgun and Viviana O. Salvadori.

Selling a Stock at the Maximum
Imagine an investor who owns a stock and wishes to sell it before
time T > 0 so as to maximize his profit. The investor has to
decide when to sell the stock. Naturally, he would like to sell
when the stock price is at its maximal value over the interval
[0,T], but such a strategy is impractical since this information
is only known at time T . What the investor would like to do
at any time t[0,T] is to use all the accumulated information to
infer how close the stock price is to the ultimate maximum, and
based on this decide whether he should sell or not. The investor
is thus faced with solving an optimal prediction problem a type
of optimal stopping problem where the gain function depends
on the future and is only FT measurable.
We will present a solution to this problem when the stock
price follows geometric Brownian motion. The optimal prediction problem naturally leads to a free boundary problem, and we
will discuss how local timespace calculus techniques from probability theory can be used to characterize the free boundary and
the value function.
This is joint work with Prof. Goran Peskir.

On the Existence of Extreme Waves and the Stokes
Conjecture With Vorticity
We present some recent results on singular solutions of the problem of traveling gravity water waves on flows with vorticity. We
show that, for a certain class of vorticity functions, a sequence
of regular waves converges to an extreme wave with stagnation
points on the free surface. We also show that, for any vorticity function, the profile of an extreme wave must have either a
corner of 120° or a horizontal tangent at any stagnation point
about which it is supposed symmetric. Moreover, the profile
necessarily has a corner of 120° if the vorticity is nonnegative
near the free surface.

LongTime Asymptotics for Some HeleShaw Models
With Injection or Suction in a Single Point
We discuss longtime behaviour of HeleShaw ow with injection
and suction in a single point, for domains that are small perturbations of balls. An evolution equation for the motion of these
domains is derived and linearised. We use spectral properties
of the linearisation of this equation to show that in the case of
injection, perturbations of balls decay algebraically. If for the
threedimensional case surface tension is included, all liquid can
be removed by suction if the suction point and the geometric
centre coincide and the ratio of suction speed and surface tension is small enough. We also show the existence of noncircular,
selfsimilar solutions up to complete extinction. These solutions
are found as bifurcation solutions to a nonlocal elliptic equation
of order three.