The Riemann Mapping Theorem tells us that planar simply
connected domains (different from the entire complex plane) are
conformally equivalent. A conformal map is a diffeomorphic solution
of the Cauchy-Riemann system. For generic multiply connected domains,
however, there is no such mapping. We are interested in solving the
Cauchy-Riemann system in the least squares sense. Such diffeomorphic
solutions minimize the Dirichlet energy and, therefore, are harmonic.
We establish the existence of such energy minimal diffeomorphisms between
doubly connected planar domains. This talk is based on joint work with
T. Iwaniec, N-T. Koh and L. Kovalev.
Pulsating traveling fronts are solutions of heterogeneous
reaction-advection-diffusion equations that model some population
dynamics. Fixing a unitary direction e, it is a well-known fact that
for nonlinearities of KPP type (after Kolmogorov, Petrovsky and
Piskunov, $f(u)=u(1-u)$ is a typical homogeneous KPP nonlinearity),
there exists a minimal speed c* such that a pulsating traveling front
with a speed c in the direction of e exists if and only if $c\geq c*$.
In a periodic heterogeneous framework we have the formula of Berestycki,
Hamel and Nadirashvili (2005) for the minimal speed of propagation.
This formula involves elliptic eigenvalue problems whose coefficients
are expressed in terms of the geometry of the domain, the direction of
propagation,
and the coefficients of reaction, diffusion and advection
of our equation. In this talk, I will describe the asymptotic behaviors
of the minimal speed of propagation within a large drift. These ``large
drift limits'' are expressed as maxima of certain variational quantities
over the family of ``first integrals'' of the advection field. I will
give more details about the limit and a necessary and sufficient condition,
in terms of the drift term, for which the regime is sharp in the 2-d case.
There have been, over the last 14 years, a number of
far reaching results generalizing the famous original
F. and M. Riesz's uniqueness theorem that states that
if a bounded analytic function in the unit disc of the
complex plane has the same radial limit in a set of
positive Lebesgue measure on its boundary, then the
function has to be constant.
In this talk we revise some of those results and give
a more general condition under which we will still be
able to show that the function is identically constant.
Let us remark here that the previous results were valid
for real valued functions, while our result is more in the
spirit of the initial results of M. and F. Riesz and
Beurling, where they considered functions from the complex
plane into the complex plane. Namely, our results hold for
functions defined in the unit ball of the n-dimensional
euclidean space into the n-dimensional euclidean space.
We consider degenerate parabolic equations and we prove that for
such kind of equations we have that a Harnack inequality holds.
This result has a by-product, a new proof of the regularity of
solutions of such kind of equations.
The classical theorem of Federer and Ziemer says that the set of
infinite value of a $W^{1,p}$ function has Hausdorff dimension at most $n-p$.
Assuming the integrability of $u^{-\alpha}$, we can get an estimate of the
size of the set of zero value. The proof of such estimate depends on a special
Poincare inequality, and is totally different from the proof of Federer-Ziemer
Theorem.
5:00-5:50pm
Thack 703
Greg Francos (University of Pittsburgh)
will present the following paper:
G. Alberti, L. Ambrosio: A geometrical approach to monotone
functions in $R^n$
Math. Z. 230 (1999), 259--316.
Jose Gonzalez Llorente (Universidad Autónoma de Barcelona)
On differentiability properties of Zygmund and Weirstrass-type functions
The borderline examples in Weierstrass construction of continous, nowhere
differentiable functions belong to the so called Zygmund class. We survey
some classical and not so classical results on the differentiability
properties of Zygmund and Weierstrass-type functions on the real line. Then
we will give some recent extensions to the higher dimensional case.
5:00-5:50pm
Thack 703
Hoang Tran (University of Pittsburgh)
will present the following paper:
W. Gangbo, An elementary proof of the polar decomposition of vector-valued
functions.
Arch. Rat. Mech. Anal., Vol. 128, 380-399, 1995.
Stanislav Hencl (Charles University in Prague)
Jacobians of Sobolev homeomorphisms
Carlos Mora-Corral (Basque Center for Applied Mathematics)
A variational model for cavitation and fracture in nonlinear elasticity
We propose a variational model for cavitation and fracture in the context
of nonlinear elasticity. The energy to minimize is the sum of the elastic energy
plus a surface energy accounting for the creation of surface. We prove the
existence of minimizers that are one-to-one and orientation-preserving. We explain
the connections with the theory of Cartesian currents and with the regularity
of inverses of weakly differentiable functions. This is a joint work with Duvan Henao.
Greg Francos (University of Pittsburgh)
The Whitney extension theorem
Pawel Goldstein (Warsaw University)
Trajectories of analytic gradient vector fields
The question about the regularity of both the flow and the
trajectories of an analytic gradient vector field first arose in the
work of Rene Thom and Stalislaw Lojasiewicz on semi- and subanalytic
sets. The restriction to gradients of analytic, instead of smooth
functions excludes phenomena like trajectories spiralling around a
critical point (a highly non-trivial fact), and it is conjectured that
in many aspects they behave like semi-analytic arcs. A list of simply
formulated problems is easily settled in dimension 2, however, very
few results exist in dimension 3 - and even less is known in higher
dimensions.
I shall present the aforementioned conjectures and sketch a complete
answer to them in the case when the function is harmonic and the
dimension is 3.
Pawel Goldstein (Warsaw University)
Trajectories of analytic gradient vector fields
The question about the regularity of both the flow and the
trajectories of an analytic gradient vector field first arose in the
work of Rene Thom and Stalislaw Lojasiewicz on semi- and subanalytic
sets. The restriction to gradients of analytic, instead of smooth
functions excludes phenomena like trajectories spiralling around a
critical point (a highly non-trivial fact), and it is conjectured that
in many aspects they behave like semi-analytic arcs. A list of simply
formulated problems is easily settled in dimension 2, however, very
few results exist in dimension 3 - and even less is known in higher
dimensions.
I shall present the aforementioned conjectures and sketch a complete
answer to them in the case when the function is harmonic and the
dimension is 3.
Mikko Parviainen (Helsinki University of Technology)
Parabolic mean value theorems and applications
In this talk, we study viscosity solutions to a class of nonlinear parabolic
equations in terms of a mean value property. We also consider stochastic games.
This talk is based on a joint work with J.J. Manfredi and J.D. Rossi.
Pawel Konieczny (University of Minnesota, IMA)
Directional approach to spatial structure of solutions to the
Navier-Stokes equations in the plane
We investigate the steady state Navier-Stokes equations considered in the
full space $R^2$. We suplement the system with a condition at infinity which
requires the solution (the velocity) to tend to a prescribed constant vector
field. This problem is strictly connected with an open problem of a flow past
an obstacle on the plane. The main difficulty there is to assure the convergence
of a solution to a prescribed velocity at infinity. We propose a new method to
deal with this problem. The class of functions, where we look for a solution, is
different from standard Sobolev spaces. This is due to the fact that our analysis
is carried through in a Fourier space only in one direction. We show existence of
solutions together with their basic asymptotic structure.
Jasun Gong (University of Pittsburgh)
Analysis on Manifolds, Part II: Geometric Inequalities
Under the assumption of lower bounds on Ricci curvature of a Riemannian
manifold, we will discuss (1) a proof of a local Poincare inequality and
(2) the interplay between Sobolev inequalities and isoperimetric inequalities.
Jasun Gong (University of Pittsburgh)
Analysis on Manifolds, Part I: Curvature Conditions
This will be the first in a series of expository talks about manifolds
with bounded curvature and geometric analysis on them. We will review
some familiar notions from Riemannian geometry and then discuss analytic
consequences of curvature conditions. In forthcoming talks, we will discuss
various well-known results in the literature, such as isoperimetric
inequalities and generalized Liouville theorems, and possibly generalizations
of these results to the metric space setting.
Harri Varpanen (University of Jyvaskyla)
On Wolff's anti-Fatou theorem for p-harmonic functions
In 1984 Tom Wolff constructed a bounded $p$-harmonic function
in the upper half-plane having radial limits almost nowhere on
the boundary of the half-plane. He also conjectured that the
construction generalizes to other domains (such as the disk).
I will describe Wolff's construction and talk about the ongoing
work of verifying the conjecture for the disk.
Piotr Haj³asz (University of Pittsburgh)
Approximation of Sobolev mappings between manifolds III
This is a continuation of the talk from the last week.
We will discuss recent results about density of smooth mappings
in the class of Sobolev mappings between manifolds. In particular we will
discuss the complete solution of the problem obtained by Hang and Lin in 2003.
Piotr Haj³asz (University of Pittsburgh)
Approximation of Sobolev mappings between manifolds II
This is a continuation of the talk from the last week.
We will discuss recent results about density of smooth mappings
in the class of Sobolev mappings between manifolds. In particular we will
discuss the complete solution of the problem obtained by Hang and Lin in 2003.
Piotr Haj³asz (University of Pittsburgh)
Approximation of Sobolev mappings between manifolds I
We will discuss recent results about density of smooth mappings
in the class of Sobolev mappings between manifolds. In particular we will
discuss the complete solution of the problem obtained by Hang and Lin in 2003.
Filippo Cagnetti (Carnegie Mellon University)
An Extension Theorem in SBV
We show the existence of an extension operator for special functions with
bounded variation with a careful energy estimate. The main application is a
compactness result for non-coercive functionals consisting of a volume and a
surface integral. More precisely, we will focus on the study of the asymptotic
behaviour of the Mumford-Shah functional on a periodically perforated domain,
as the size of the holes and the periodicity parameter of the structure tend to
zero.
Pablo Pedregal Tercero (Universidad de Castilla-La Mancha, Spain)
A variational approach to ODE and its numerical implementation
We will describe an optimization approach to the analysis of ODE based on
minimizing an error functional. This minimization process requires some
non-standard ingredients but its analysis is quite elementary. After showing
existence and uniqueness, we will focus on a numerical implementation
based on this approach, and test it with several typical examples.
Marta Lewicka (Carnegie Mellon University and University of Minnesota)
The matching property of infinitesimal isometries on elliptic surfaces
and elasticity of thin shells
A central problem in the mathematical theory of elasticity is to predict
theories of
lower-dimensional objects subject to mechanical deformations,
starting from the 3d nonlinear theory. For shells, despite extensive use of
the ad-hoc generalizations of plate theories present in the engineering
applications, not much is known from the mathematical point of view.
In a recent effort, the limiting behavior (using the notion of Gamma-limit)
of the 3d nonlinear elasticity for thin elliptic shells has been described,
as the shell thickness h converges to 0, and under the assumption that the
elastic energy of deformations scales like $h^\beta$, with $\beta>2$.
In this talk I will mainly concentrate on explaining the two major ingredients
of the proofs, which are: the density of smooth maps in the space of Sobolev
first order isometries, and a result on matching smooth infinitesimal isometries
to exact isometric immersions.
This is joint work with Maria Giovanna Mora (SISSA) and
Reza Pakzad (University of Pittsburgh).
5:00-5:50 pm
Thack 703
Pawel Konieczny (Carnegie Mellon University)
Thorough analysis of the Oseen system in 2D exterior domains
We construct $L^p$-estimates for the inhomogeneous stationary Oseen
system studied in a two dimensional exterior domain with inhomogeneous
slip boundary conditions. The main part of the talk is a presentation
of results for the half space $\mathbb{R}^2_+$, which are substantial for
the exterior problem. Main tools are given by the Fourier analysis in
order to obtain maximal regularity estimates. In addition, these optimal
estimates show us a difference between points on the boundary in front
of the obstacle and behind the obstacle. The former are typical for elliptic
problems while the latter show disturbance which is typical for parabolic
problems.
Marco Barchiesi (Carnegie Mellon University)
New counterexamples to the cell formula in nonconvex homogenization
I will show that for the homogenization of multiple integrals, the
quasiconvexification of the cell formula is different from the asymptotic
formula in general. To this aim, I will construct two examples in different
settings: the homogenization of a composite material and the
homogenization of a homogeneous material on a perforated domain.
This is a joint work with A. Gloria.
Jose' Miguel Urbano
(CMUC, University of Coimbra, Portugal)
Entropy solutions for nonlinear elliptic problems with variable growth and L^1 data
Taking as a model the $p(x)$-Laplace equation, we extend the theory of entropy
solutions to elliptic equations with nonlinearities involving variable
exponents. We obtain existence and uniqueness for $L^1$ data, as well as
integrability results for the solution and its gradient. The obstacle
problem is also studied: we prove some convergence and stability properties
of the coincidence set, extending the Lewy-Stampacchia inequalities to the
general framework of $L^1$. This is a topic where PDEs again meet Functional
Analysis in a truly two-way street and we make a brief tour of the properties
of functional spaces involving variable exponents. For Marcinkiewicz
spaces, we present new inclusions of independent interest.
5:00-5:50 pm
Thack 703
Zhuomin Liu (University of Pittsburgh)
A compact embedding for Sobolev spaces is equivalent to a better embedding
If a Sobolev space $W^{1,p}$ on some bounded domain is embedded into $L^q$,
then it is well known that for any $r$ less than $q$ the embedding into $L^r$
is compact. The same holds for embeddings of $W^{1,p}$ into Orlicz spaces.
If a Sobolev spaces is embedded into an Orlicz space with a given
N-function A, then for Orlicz spaces with the N-functions that increare
essentially more slowly at infinity than A, the embedding is compact.
In the talk I will show that in the presence of the $\Delta_2$ condition
the converse implication is also true. Compact embedding implies the existence
of a bounded embedding into a better Orlicz space.
This is a joint work with Piotr Hajlasz.
Antonio Ferriz (University of Vigo (Spain))
Estimating how hot it is at the center of the Sun
Calculations at undergraduate level can be used to show how to estimate
the pressure and the temperature at the center of the Sun
(under the assumption that the Sun is a sphere of gas in hydrostatic
equilibrium). A systematization of the procedure leads to the
"virial theorem", which allows an insight into other interesting aspects
of the Sun as a selfgravitating body: the Kelvin-Helmholtz time scale,
the basic oscillation period, or the concept of "negative" heat capacity.
Within this framework I will review nineteenth-century attempts to estimate
the age of the Sun and to understand its energy source.
Tuomo Kuusi (Columbia University/Helsinki University of Technology)
Harnack estimates for weak supersolutions to nonlinear
degenerate parabolic equations
We give an overview of a proof for a parabolic weak Harnack estimate for weak
supersolutions to nonlinear degenerate parabolic equations. In the quasilinear
case, Moser's classical proof for Harnack's inequality for solutions uses
the Harnack estimate for supersolutions and reverse Hölder’s inequality
for subsolutions. It turns out that our result gives a similar proof in
the nonlinear case.
5:00-5:50 pm
Thack 703
Jasun Gong (University of Pittsburgh)
On Null Sets and Derivations in the Plane
In 2005, Alberti, Csornyei, and Preiss proved a new covering theorem
for Lebesgue null sets in the plane. As a consequence of this fact,
we will discuss how Lebesgue singular measures in the plane cannot
admit rank-2 modules of derivations. We will review Weaver's theory of
derivations and relevant facts as necessary.
Patrick Rabier (University of Pittsburgh)
Bifurcation of periodic solutions in nonlinear evolution
problems with periodic forcing
5:00-5:50 pm
Thack 703
Jasun Gong (University of Pittsburgh)
Derivations on Metric Measure Spaces
Following the work of N. Weaver, we will discuss a theory of
derivations. Such structures determine a generalized first-order calculus
on metric spaces equipped with a measure. As we will see, derivations
have many properties that are similar to vector fields on smooth
manifolds. In this talk we will present these facts, as well as new results
about derivations in the setting of Euclidean spaces.
Juan Manfredi (University of Pittsburgh)
Mean Value Properties
We characterize p-harmonic functions using mean value properties that hold
asymptotically. The results are interesting even for the well-known case $p=2$.
Mikko Parviainen (University of Pittsburgh/Helsinki University of Technology)
Parabolic quasiminimizers and higher integrability
Quasiminimizers provide a unifying approach in the calculus of
variations but also create new mathematical theory. It turns out
that a parabolic quasiminimizer globally belongs to a higher Sobolev
space than assumed a priori. This can be deduced from the fact that the
gradient satisfies a reverse Hölder inequality near the boundary. In this
talk, we discuss the parabolic quasiminimizers and touch some aspects of
the higher interability proof.
Jasun Gong (University of Pittsburgh)
An Extension Theorem for Quasiconformal Mappings on Spheres II
(after F. Gehring)
Cancelled
Jasun Gong (University of Pittsburgh)
An Extension Theorem for Quasiconformal Mappings on Spheres
(after F. Gehring)
In 1970, F. Gehring showed that any quasi-conformal homeomorphism between
"collared" domains (in Euclidean spaces of any dimension) extends to
a quasi-conformal homeomorphism of the resulting Jordan domains. We will
discuss a sketch of his proof and review relevant facts as needed.