


Nonlinear Analysis
Org: Martial Agueh (Victoria), Ivar Ekeland (PIMS) and Robert McCann (Toronto) [PDF]
 RUSTUM CHOKSI, Simon Fraser University
The Periodic Isoperimetric and CahnHilliard Problems and
Nonlocal Perturbations
[PDF] 
I will discuss two classical problems in analysis and geometry posed
on ndimensional flat torus. I will focus on some interesting
questions which remain open. I will then discuss a nonlocal
isoperimetric problem with an emphasis on the calculation and
application of its first and second variations.
This talk is based on joint work with Peter Sternberg (Indiana
University).
 DANIELA DE SILVA, MSRI, 17 Gauss Way, Berkeley, CA 94720, USA
A singular energy minimizing free boundary
[PDF] 
In this talk, we consider the problem of minimizing the energy
functional ò(Ñu^{2} + c_{{u > 0}}). We will exhibit
the first example of a singular energy minimizer, which occurs in
dimension n=7. This is the analogue of the 8dimensional Simons
cone in the theory of minimal surfaces.
This is joint work with D. Jerison.
 IVAR EKELAND, University of British Columbia
A new type of equation arising from noncommitment in
economics
[PDF] 
In control theory one traditionally uses an exponential discount
factor to arbitrage future gains agains current losses. It is well
known that the optimal control problem then is solved by a
HamiltonJacobiBellman equation for the value function. In joint
work with Ali Lazrak, we point out that this approach completely
breaks down when the discount is not exponential, and the
decisionmaker cannot commit. Instead of looking for optimal
controls, one must then look for equilibrium strategies, and the HJB
equation is then replaced by a remarkable integrodifferential
equation.
 PIERPAOLO ESPOSITO, UBC, PIMS, 1933 West Mall, Vancouver, BC V6T 1Z2
On a semilinear PDE with a singular nonlinearity
[PDF] 
I will discuss some recent progress on the semilinear elliptic problem
Du = [(lf(x))/((1+u)^{2})] on a smooth bounded domain
W of R^{N} with an homogeneous Dirichlet boundary
condition. This equation models a simple electrostatic
MicroElectromechanical System (MEMS) device and has been studied
recently by Pelesco, and by GuoPanWard. Guo and Ghoussoub
showamong other thingsthat the branch of minimal solutions
u_{l} is compact up to a certain critical value l^{*},
provided 1 £ N £ 7. In this talk, I will describe an analogous
result for the second branch (of "mountain pass" solutions), which
holds in the same low dimensions. Our techniques rely on a careful
blowup analysis for solutions satisfying certain spectral properties.
Joint work with N. Ghoussoub and Y. Guo.
 RICHARD FROESE, University of British Columbia
AC spectrum for the Anderson model on a tree: a geometric proof
[PDF] 
We give a new proof of Klein's result on the existence of absolutely
continuous spectrum for a discrete random Schrödinger operator on a
tree with small disorder. Our proof relies on a new geometric way of
controlling the Green's function, based on the contraction properties
of a transformation in hyperbolic space.
This is joint work with David Hasler and Wolfgang Spitzer.
 NASSIF GHOUSSOUB, University of British Columbia
Concentration estimates for EmdenFowler equations with
boundary singularities and critical growth
[PDF] 
We establish existence and multiplicity of solutions for the Dirichlet
problem å_{i} ¶_{ii} u + [(u^{2*2}u)/(x^{s})] = 0 on
smooth bounded domains W of R^{n} (n ³ 3) involving the
critical HardySobolev exponent 2^{*}
= [(2(ns))/(n2)] where
0 < s < 2, and in the case where zero (the point of singularity) is on
the boundary ¶W. Just as in the Yamabetype
nonsingular framework (i.e., when s=0), there is no
nontrivial solution under global convexity assumption (e.g.,
when W is starshaped around 0). However, in contrast to the
nonsatisfactory situation of the nonsingular case, we show the
existence of an infinite number of solutions under an assumption of
local strict concavity of ¶W at 0 in at least one
direction. More precisely, we need the principal curvatures of
¶W at 0 to be nonpositive but not all vanishing. We
also show that the best constant in the HardySobolev inequality is
attained as long as the mean curvature of ¶W at 0 is
negative. The key ingredients in our proof are refined concentration
estimates which yield compactness for certain PalaisSmale sequences
which do not hold in the nonsingular case.
This is joint work with Frederic Robert.
 STEPHEN GUSTAFSON, UBC Dept. of Mathematics, 1984 Mathematics Rd., Vancouver,
BC V6T 1Z2
Schroedinger maps near harmonic maps
[PDF] 
The Schroedinger map equation is a basic model in ferromagnetism, as
well as a geometric (and hence nonlinear) version of the linear
Schroedinger equation. An important open question is whether finite
energy solutions are globally smooth, or blow up in finite time. We
describe some results for equivariant Schroedinger maps from
2+1dimensional spacetime into the 2sphere, with energy close to
the energy of harmonic maps.
This is joint work with Kyungkeun Kang and TaiPeng Tsai.
 DAVID HARTENSTINE, Western Washington University
Regularity of Weak Solutions of the MongeAmpère Equation
[PDF] 
Regularity properties of Aleksandrov solutions to the Dirichlet
problem for the MongeAmpère equation detD^{2} u = m where
m is a Borel measure on a convex domain in R^{n} will be
discussed. The measure m satisfies a condition, introduced by
Jerison, that is weaker than the doubling condition. Some of the
results of Caffarelli's regularity theory for the MongeAmpère
equation, more specifically strict convexity and interior
C^{1,a} regularity, are extended to the solutions of these
problems.
This is joint work with Cristian Gutierrez.
 RICHARD KENYON, UBC, Vancouver
Dimers and the complex Burgers equation
[PDF] 
We study a simple model of crystalline surfaces in R^{3}.
Microscopically, these are random discrete surfaces, arising in the
socalled dimer model, or domino tiling model. The law of large
numbers implies that at large scales the surfaces take on definite
shapes, which are smooth surfaces satisfying a certain PDE, related to
the complex Burgers equation. We show how this equation can be solved
via complex analytic functions, and investigate the behavior of
solutions, in particular the formation of facets. This is the first
model of facet formation which can be analytically solved.
This is joint work with Andrei Okounkov.
 RUEDIGER LANDES, University of Oklahoma, Norman, OK 73071
Wave front solutions in the theory of boiling liquids
[PDF] 
In an new attempt to model the phase transition from nucleate to
transient boiling, Professor Marquardt from Aachen proposed to
consider the the heat flow in the heating vessel rather than in the
boiling liquid. In this model the heat flow in the wall of the
vessel, subject to heat equation, is combined with a nonlinear Neumann
boundary condition at the surface of the wall toward the boiling
liquid. The nonlinearity is determined by the change of the heat
conduction coefficient in the phase transition.
For the onedimensional heat equation with a nonlinear inhomogeneous
term, the existence of wavefront solutions is well known and widely
used to model phase transitions. Work of Aronson and Weinberger also
dealt with the moredimensional situation and showed that there are
subsolutions which behave like wavefronts. Consequently the actual
solutions must have a sudden change of state, also. However, this
model with the nonlinearity in the equation rather then the boundary
condition can be justified in our case for (infinitely) thin surfaces
only.
Here we present an approach which provides a wavefront type
subsolution for the nonlinear Neumann problem and hence establish a
first mathematical confirmation of Marquardt's model.
Further research will concentrate on the discussion of initial
configurations (dry spots) which generate a wavefront type solutions
and those which do not. In addition we intend to address the question
how the maximum and minimum speed of the traveling wave is determined
by the initial configuration and the other parameters of the data.
 ABBAS MOAMENI, University of British Columbia
Selfdual variational principles for periodic solutions of
Hamiltonian and other dynamical systems
[PDF] 
Selfdual variational principles are introduced in order to construct
solutions for Hamiltonian and other dynamical systems which satisfy a
variety of linear and nonlinear boundary conditions including many of
the standard ones. These principles lead to new variational proofs of
the existence of parabolic flows with prescribed initial conditions,
as well as periodic, antiperiodic and skewperiodic orbits of
Hamiltonian systems.
This is a joint work with N. Ghoussoub.
 TRUYEN NGUYEN, Mathematical Sciences Research Institute, 17 Gauss Way,
Berkeley, CA 947205070
On MongeAmpère Type Equations Arising In Optimal
Transportation Problems
[PDF] 
In this talk, we will discuss MongeAmpère type equations arising
in optimal transportation problems. We prove the comparison principle,
maximum principle and also a quantitative estimate of Aleksandrov type
for cconvex functions. These results are in turn used to prove the
solvability and uniqueness of weak solutions for the Dirichlet
problems.
 ADAM OBERMAN, SFU
Numerical methods for Mass Transportation
[PDF] 
Numerical results for the mass tranportation problem will be
presented. The transportation problem with cost function which depend
on the difference xy will be considered. We approximate measures
by atoms, and project to a finite dimensional linear programming
problem, which can then be solved by standard methods.
Numerical results for linear, quadratic, and square root of distance
costs will be presented.
We will also present a convergent finite difference method for solving
the Dirichlet problem for the MongeAmpère equation.
 GUNTHER UHLMANN, University of Washington
The Calderon problem with partial data
[PDF] 
We will consider in this talk the inverse problem of determining the
electrical conductivity inside a medium by making voltage and current
measurements on subsets of the boundary. We will state and outline
the proofs of some recent results on this problem.
 MICHAEL WARD, Dept. of Mathematics, University of British Columbia,
Vancouver, BC
Eigenvalue Optimization, Spikes, and the Neumann Green's Function
[PDF] 
An optimization problem for the fundamental eigenvalue of the
Laplacian in a planar simplyconnected domain that contains N small
identicallyshaped holes, each of a small radius e << 1, is
considered. The boundary condition on the domain is assumed to be of
Neumann type, and a Dirichlet condition is imposed on the boundary of
each of the holes. The reciprocal of this eigenvalue is proportional
to the expected lifetime for Brownian motion in a domain with a
reflecting boundary that contains N small traps. For small hole
radii e, we derive an asymptotic expansion for this
eigenvalue in terms of the hole locations and the Neumann Green's
function for the Laplacian. For the unit disk, ringtype
configurations of holes are constructed to optimize the eigenvalue
with respect to the hole locations. For a onehole configuration, the
uniqueness of the optimizing hole location in symmetric and asymmetric
dumbbellshaped domains is investigated. This eigenvalue optimization
problem is shown to be closely related to the problem of determining
certain vortex configurations in the GinzburgLandau theory of
superconuctivity and to the problem of determining equilibrium
locations of particlelike solutions, called spikes, to certain
singularly perturbed nonlinear reactiondiffusion systems. Some
results for the equilibria, stability, and bifurcation behavior, of
these spike solutions are given.

