PDE Seminar, Spring 2008
Wednesday's from 4 to 5 PM at MIT (room 2-135).
Next seminar:
May 14: Eugen Varvaruca (University of Bath)
``On the existence of extreme waves and the Stokes conjecture with vorticity.''
Abstract:
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 at its crests. We also show that,
for any vorticity function, the profile of an extreme wave must have either a corner of
120o or a horizontal tangent at any stagnation point about which it is supposed symmetric.
Moreover, the profile necessarily has a corner of
120o if the vorticity is nonnegative near the free surface.
Full schedule: (Some slides/notes available below)
February 1: Yonatan Sivan (Tel Aviv University)
``Dynamics of solitons in inhomogeneous media.''
Special date and time: Friday 4 PM in 2-135
Abstract:
We study soliton dynamics in a general inhomogeneous nonlinear media. We show that the dynamics can be
predicted based on which of two stability conditions is violated: violation of the slope (Vakhitov--Kolokolov)
condition leads to a width instability, whereas failure to satisfy the spectral condition leads to a drift
instability. In addition, we show the importance of a quantitative study of the stability for predicting
the dynamics. We demonstrate our results in a variety of nonlinear media and inhomogeneities.
February 13: Gilles Angelsberg (Stanford)
``Large solutions for biharmonic maps in four dimensions.''
Abstract:
We investigate the existence of large solutions for biharmonic maps from a 4-dimensional Euclidean domain
Ω into S4.
Introducing the notion of topological degree for Sobolev maps from R4
to S4, we show that there exists locally minimizing extrinsic biharmonic
maps u* of topological degree -1 and 1. The proof is based on P.L. Lions'
concentration compactness priniciple. This allows us to exclude the phenomena of concentration and
vanishing at infinity, for minimizing sequences for the Hessian energy with prescribed topological degree
-1 or 1, up to rescalings and translations. We infer that the degree is preserved in the
limit. Then, for Ω = B1 unit ball in R4, we
show the existence of two non homotopic biharmonic maps for certain Dirichlet boundary data. The key step
is a "sphere attaching lemma" stating the existence of a map u, non homotopic to the
absolute minimizer u of the Dirichlet problem, having less energy than the sum of the energies
of u and u*. Thus, we can exclude bubbling of minimizing sequences
in the considered homotopy class in order to conclude compactness.
February 15: Lionel Levine (Berkeley)
``Free boundary problems arising from combinatorial and probabilistic growth models.''
Special date and time: Friday 4 PM in 2-135
Abstract:
Start with n particles at each of k points in the integer lattice
Zd, and let each particle perform simple random walk until it reaches
an unoccupied site. The distribution of the resulting random set of occupied sites does not depend on the
order in which the walks are performed. We prove that if the distances between the starting points are
scaled by n1/d, the set of occupied sites has a deterministic scaling limit in
Rd. The limit is identified using the "odometer" of the process,
which solves a free boundary obstacle problem for the Laplacian. The limiting shape can also be described
in terms of quadrature identities for harmonic functions, revealing an underlying connection with classical
topics in potential theory and fluid mechanics. Joint work with Yuval Peres.
February 20: Frank Merle (Université de Cergy-Pontoise)
``On collision of solitary waves for non integrable GKdV.''
February 27: Emanuel Milman (IAS)
``Quantitative stability of spectral-gap and isoperimetry under perturbations of convex domains.''
February 29: Luis Silvestre (Courant)
``Fully nonlinear integro-differential equations."
Special date and time: Friday 4 PM in 2-135
Abstract:
We study nonlinear integro-differential equations. Typical examples
are the ones that arise from stochastic control problems with
discontinuous Levy processes. We can think of these as nonlinear
equations of fractional order. Indeed, second order elliptic PDEs are
limit cases for integro-differential equations. Our aim is to extend
the theory of fully nonlinear elliptic equations to this class of
equations. We are able to obtain a result analogous to the Alexandroff
estimate, Harnack inequality and C1,α regularity. As the
order of the equation approaches two, in the limit our estimates
become the usual regularity estimates for second order elliptic pdes.
This is a joint work with Luis Caffarelli.
March 5: Yaron Ostrover (MIT)
``A Brunn-Minkowski-type inequality in Hamiltonian dynamics.''
Abstract:
The Brunn-Minkowski inequality for volumes of bodies is
a fundamental result which has numerous applications in geometry and analysis.
In this talk we discuss an analogue of this inequality for a certain
symplectic quantity given by the minimal symplectic area (action) of a
periodic orbit on a convex energy surface.
(This is a joint work with Shiri Artstein-Avidan).
March 12: Will Gryc (Morehouse College)
``On the Holonomy of the Coulomb Connection over Manifolds with Boundary.''
Abstract:
In Yang-Mills theory, one considers connections on a principal bundle π :P → M
whose structure group K is a compact subgroup of SO(m) or
SU(m). A connection A is called a Yang-Mills connection if it satisfies
the Yang-Mills equation
(dA)*ΩA = 0,
where ΩA is the curvature of A and (dA)*
is the adjoint of the exterior derivative dA induced by the connection A.
This equation is nonlinear, and Yang-Mills connections are preserved under a group of so-called
gauge transformations. This ``gauge invariance'' implies nonuniqueness of solutions
to the Yang-Mills equation, and also motivates consideration of the quotient
A→ A/ G,
where A is a suitable space of connections, and G is the corresponding
gauge group. In many circumstances, this quotient is an infinite dimensional principal bundle in its own
right, and thus allowing one to consider connections on the bundle
A→ A/ G.
One such connection is called the Coulomb connection.
Narasimhan and Ramadas showed that the restricted holonomy group of the Coulomb connection is dense in the
connected component of the identity of the gauge group when one considers the product principal bundle
S3 × SU(2) → S3. Instead of a base manifold S3,
we consider a base manifold of dimension n ≥ 2 with a boundary and use
Dirichlet boundary conditions on connections as defined by Marini. In contrast to
an intermediate result of Narasimhan and Ramadas, the image of the curvature form of the Coulomb connection
at one fixed point is never dense in the gauge algebra in this boundary-condition case. In fact, it lies in
a kernel of a linear operator which involves the mean curvature of the boundary of the manifold M.
However, if the base manifold is a bounded open subset O of ℜn with smooth
boundary and K is a compact subgroup of SO(m) or SU(m), then the restricted
holonomy group of the Coulomb connection on the product principal bundle
cl(O) × K → cl(O) is again a dense subset of the connected component of the identity
of the gauge group.
March 19: Walter Strauss (Brown)
``Steady Water Waves.''
April 9: Jill Pipher (Brown)
``Absolute continuity of elliptic measure.''
Canceled
Abstract:
In this talk we survey some methods for deriving
quantitative absolute continuity of the measure
associated to an elliptic second order operator in divergence
or nondivergence form. In particular, we are interested in
methods which work in the absence of L2 identities, and
yet give solvability of some Dirichlet or Neumann boundary
value problems. I'll discuss applications to specific operators
whose coefficients are Carleson measures.
April 16: Nam Le (Courant)
``On the convergence of critical points of the Ambrosio-Tortorelli functional.''
Abstract:
We investigate critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero
Dirichlet boundary conditions replace the fidelity term. That Dirichlet variant is the natural functional
when addressing a problem of brittle fracture in an elastic material. We show that, in the one-dimensional
case, they converge to specific critical points of the corresponding variant of the Mumford-Shah functional.
These critical points exhibit many symmetries. Conversely, given any symmetric critical point of the
Mumford-Shah functional, we prove that it is actually a limit of critical points of the Ambrosio-Tortorelli
functional. This is a joint work with Gilles Francfort and Sylvia Serfaty.
April 23: Andrej Zlatos (Chicago)
``Speed-up of reaction-diffusion fronts by flows.''
Abstract:
I will discuss some recent results on speed-up of traveling fronts by strong periodic flows in
reaction-diffusion equations. I will present a characterization of the flows which can arbitrarily
speed up fronts for general combustion-type reactions in two dimensions. The rate of this speed-up
will also be determined. The problem turns out to be closely connected to the simpler question of
effective diffusivity enhancement in the homogenization of the corresponding (linear) passive scalar
equations.
April 30: Alessio Figalli (Nice)
``Quantitative isoperimetric inequalities and optimal transport.''
Special time and place: 2 PM in 4-149
Abstract:
In this talk I will show how one can prove a sharp quantitative version of the
anisotropic isoperimetric inequality by exploiting mass transportation theory,
especially Gromov's proof of the isoperimetric inequality and the Brenier-McCann
Theorem.
This is a joint work with F. Maggi and A. Pratelli.
May 7: Mathieu Lewin (CNRS & Cergy-Pontoise)
``The thermodynamic limit of quantum Coulomb systems.''
Abstract:
I will present a new approach for proving the existence of the thermodynamic
limit for quantum systems composed of electrons and nuclei interacting via
the Coulomb potential, like in ordinary matter. In particular I will provide
a very general setting allowing to study many different quantum systems.
This is a joint work with Christian Hainzl and Jan Philip Solovej.
May 14: Eugen Varvaruca (University of Bath)
``On the existence of extreme waves and the Stokes conjecture with vorticity.''
Abstract:
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 at its crests. We also show that,
for any vorticity function, the profile of an extreme wave must have either a corner of
120o or a horizontal tangent at any stagnation point about which it is supposed symmetric.
Moreover, the profile necessarily has a corner of
120o if the vorticity is nonnegative near the free surface.
Previous semesters: (Some slides/notes available below)
Fall Semester 2007
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