Title:  Using the method of layer potentials to solve a mixed boundary value problem

Abstract:  Following the exposition by William McLean in his book Strongly Elliptic Systems and Boundary Integral Equations, we use the method of layer potentials to show that on a bounded Lipschitz domain, the mixed problem for Laplace’s equation is equivalent to a 2 × 2 system of boundary integral equations.

Title: Self-improvement properties for nonlocal equations

Abstract:  I will present some results related to generalization of Meyers result to nonlocal equation. It happens that any weak solution of a nonlocal equation with data in L2 is automatically better at the integrability AND differentiability scale. This is a completely new phenomenon relying on the nonlocality of the operator. The proof is based on a new stopping time argument and a suitable generalization of Gehring lemma.

Title:  Augmented eigenfucntions: a new spectral object appearing in the integral representation of the solution of linear initial-boundary value problems.

Abstract:  We study initial-boundary value problems for linear, constant-coefficient partial differential equations of arbitrary order, on a finite or semi-infinite domain, with arbitrary boundary conditions. It has been shown that the recent Unified Transform Method of Fokas can be used to solve all such classically well-posed problems. The solution thus obtained is expressed as an integral, which represents a new kind of spectral transform. We compare the new method, and its solution representation, with classical Fourier transform techniques, and the resulting solution representation. In doing so, we discover a new species of spectral object, encoded by the spectral transforms of the Unified Method.

Title:  Extremal functions in modules of systems of measures

Abstract:  We study Fuglede’s p-modules of systems of measures in condensers in the Euclidean spaces. First, we generalize the result by Rodin that provides a way to compute the extremal function and the 2-module of a family of curves in the plane to a variety of other settings. More specifically, in the Euclidean space we compute the p-module of images of families of connecting curves and families of separating sets with respect to the plates of a condenser under homeomorphisms with some assumed regularity. Then we calculate the module and find the extremal measures for the spherical ring domain on polarizable Carnot groups and extend Rodin’s theorem to the spherical ring domain on the Heisenberg group. Applications to special functions and examples will be provided. Joint work with Melkana Brakalova and Irina Markina.

Title:  On a generalized Derivative Nonlinear SchrÖdinger equation

Abstract:  The Derivative Nonlinear SchrÖdinger equation (DNLS) equation iψt + ψxx + i |ψ|2 ψx = 0 is a canonical equation obtained from the Hall-MHD equations in a long-wave scaling, in the context of weakly nonlinear Alfvén waves propagating along an ambient magnetic field. It has the same scaling properties as the Nonlinear SchrÖdinger equation with quantic power law nonlinearity (L2-critical) that develop singularities in a finite time.  It also has the property of being completely integrable by the inverse scattering transform and has soliton solutions. In an effort to address the open question of long-time existence, we introduced recently an L2 -supercritical version of the DNLS equation by modifying the nonlinearity |ψ|2ψx to |ψ|2σψx (σ > 1). Numerical simulations indicate that a finite time singularity may occur, and provide a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile. The (complex valued) profile satisfies a nonlinear elliptic equation Qξξ −Q+ia(1/2σQ+ξQξ) − ibQξ + i |Q|2σQξ = 0, where the (real) coefficients a and b depend on σ (but not on the initial condition). Using methods of asymptotic analysis, we study the deformation of the functions Q, and parameters a, b as the nonlinearity σ tends to 1. We also check our analysis against a numerical integration of the profile equation with continuation type methods. This is an ongoing work with G. Simpson and Y. Cher.

Title:  On rotating star solutions to the Euler-Poisson equations

Abstract:  The Euler-Poisson equations are used in astrophysics to model the motion of gaseous stars. The so called rotating star solutions are density functions that satisfy the Euler- Poisson equations with a prescribed angular velocity configuration. They are one of the many efforts to try to characterize the equilibrium shape of fluids under self gravitation. Auchmuty and Beals in 1971 found a family of rotating star solutions by solving a variational free boundary problem. Recent interests in the astrophysics community require one to extend the picture to include a solid core together with its gravitational fields. In this talk, we will discuss an extension of the Auchmuty and Beals result in this direction. If time permits, we will also explore results on non-existence of solutions for fast rotation, and discuss the effects of gas equation of state.

Title:  A scattering map in two dimensions

Abstract:  We consider the scattering map introduced by Beals and Coifman and Fokas and Ablowitz that may be used to transform one of the Davey-Stewartson equations to a linear evolution. We give mapping properties of this map on weighted L 2 Sobolev spaces that mimic well-known properties of the Fourier transform. This is joint work with N. Serpico, P. Perry and K. Ott.

Title:  Uniform estimates in homogenization and applications

Abstract:  In a seminal paper of 1987, M. Avellaneda and F.H. Lin have introduced a powerfull method to show uniform Hölder and Lipschitz estimates for elliptic systems with oscillating coefficients. In this talk, I will investigate some consequences of these estimates for the large scale behavior of potentials and the asymptotics of boundary layers in homogenization. I will also address a generalization of the Lipschitz estimate to domains with oscillating boundary. The latter is a joint work with C. Kenig.

Title:  Eigenvalue Multiplicities of the Hodge Laplacian on Coexact 2-Forms for Generic Metrics on 5-Manifolds

Abstract:  In 1976, Uhlenbeck used transversality theory to show that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator are all simple for a residual set of C^r metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of C^r metrics such that the nonzero eigenvalues of the Hodge Laplacian on k forms are all simple.  We continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms by proving that for a residual set of C^r metrics, the nonzero eigenvalues of the Hodge Laplacian acting on coexact 2-forms on a closed 5-manifold have multiplicity 2.  We structure our argument around a study of the Beltrami operator, using techniques from perturbation theory to show that the Beltrami operator has only simple eigenvalues for a residual set of metrics.  We further establish even eigenvalue multiplicities for the Hodge Laplacian acting on coexact k-forms in the more general setting n=4m+1 and k=2m.

Title:  DeBranges' Proof of the Bieberbach Conjecture

Abstract:  See Bulletin Board on 7th Floor- POT for abstract