## Department of Mathematics, University of Kentucky PDE Seminar

Fall 2008

Date SPEAKER and TITLE Host
September 9 Professor John Lewis, University of Kentucky
TITLE: Where we are at with $p$ harmonic functions
Changyou Wang
September 16 Professor Peter Hislop, University of Kentucky
TITLE: Resonances for Schr\"odinger operators with compactly supported potentials
ABSTRACT I will discuss estimates on the resonance counting function for Schr\"odinger operators $H_V = - \Delta + V$ on $L^2 ( \R^d)$, for $d \geq 1$, with generic, compactly-supported, real- or complex-valued potentials $V$. The main result is that these functions have the maximal order of growth $d$. For the even dimensional case, it is shown that the functions have maximal order of growth on each sheet $\Lambda_m$, $m \in \Z \backslash \{ 0 \}$, of the logarithmic Riemann surface. We obtain this result by constructing a certain plurisubharmonic function from the determinant of a modified $S$-matrix and proving that the order of growth of the counting function can be recovered from a suitable estimate on this function. We also construct an example in each dimension of such a potential having a resonance counting function bounded below by $C_m r^d$ on each sheet $\Lambda_m$, $m \in \Z \backslash \{0\}$. this is joint work with T. Christiansen.
Changyou Wang
September 23 Professor Wu, Zhijian, University of Alabama
TITLE: Area Operators from $\mathcal{H}^p$ Spaces to $L^q$ Spaces
ABSTRACT We characterize non-negative measures $\mu$ on the unit disk $\mathbb{D}$ for which the area operator $A_{\mu}$ is bounded or compact from Hardy space $\mathcal{H}^p$ to $L^q\left(\partial\mathbb{D}\right)$ spaces.
Zhongwei Shen
September 30 Joel Kilty, University of Kentucky
TITLE: The $L^p$ Regularity and Dirichlet problems for second order elliptic PDE
ABSTRACT : In this talk we will prove that the solvability of the $L^p$ regularity problem for second order elliptic PDE is equivalent to the solvability of the $L^{p'}$ Dirichlet problem. This is joint work with Dr. Zhongwei Shen.
Zhongwei Shen
October 7 Dr. Katharine Ott, University of Kentucky
TITLE: The spectral radius conjecture on Besov spaces
ABSTRACT. In this talk we aim to prove that the spectral radius of the boundary version of the double layer potential operator, $K_{Laplace}$ or $K_{Lame}$, acting on homogeneous Besov spaces $\dot{B}^{p,q}_{s}$ is less than 1 for certain values of $p, s$. This is (ongoing) joint work with Irinia Mitrea and Tunde Jakab.
Changyou Wang
October 14 Professor Misha Feldman, University of Wisconsin, Madison
TITLE: Shock reflection, free boundary problems, and degenerate elliptic equations
ABSTRACT. In this talk we will start with discussion of shock reflection phenomena. Then we describe some recent results on existence and regularity of global solutions to shock reflection for potential flow, and discuss the techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, with ellipticity degenerate near a part of the boundary (the sonic line). We will discuss techniques to handle such free boundary problems and degenerate elliptic equations. This is a joint work with Gui-Qiang Chen.
Changyou Wang
October 21 Zhongyi Nie, University of Kentucky
TITLE: The Estimate For A Class of Multi-linear Functional
ABSTRACT : An estimate for a class of multi-linear functional is established by multi-linear interpolation methods. The result can be used to estimate the scattering transform of the Davey-Stewartson system, which allows us to assert that for small initial data, the solution of the Davey-Stewartson system depends continuously on the initial data in L2 norm.
Russell Brown
October 28 Professor Xu, Deliang, Shanghai Jiaotong University, China/University of Kentucky
TITLE: Analysis of Dirac-harmonic maps
ABSTRACT : In this talk, I will introduce the notion of Dirac-harmonic maps from a spin manifold to another target manifold. I will present some results concerning the regularity of weakly Dirac-harmonic maps from a spin Riemann surface and stationary Dirac-harmonic maps in higher dimensions, and weak compactness theorem. The method involves moving frame, Uhlenbeck-Riviere's Coulomb gauge construction, the duality between Hardy and BMO spaces, and Morrey space estimates. This is a joint work with Changyou Wang.
Changyou Wang
October 31 Professor Andy Vogel, Syracuse University
TITLE: p-harmonic measure in space
ABSTRACT : The results of Lewis and Nystr\"om on positive p-harmonic functions in a domain in $\mathbb{R}^n$ vanishing continuously on a portion of the boundary of the domain provide us with the tools we need to carry out Tom Wolff's snowflake construction for p-harmonic measures in $\mathbb{R}^n$, $n\geq 3$. For $p\neq 2$ the results of the construction are actually simpler (remarkably different from the $p=2$ case), while the arguments involved are more complicated. In particular, for any choice of a "blip" in the construction, as long as it is flat enough, the dimension of p-harmonic measure is $p>2$ while the dimension of p-harmonic measure is $>n-1$ for $10$ small enough. This is joint work in progress with Bennewitz, Lewis and Nystrom.
John Lewis
November 4 Election day, No seminar TBA
November 11 Professor Gerardo Mendoza, Temple University
TITLE : Elliptic operators on manifolds with conical singularities
ABSTRACT : The principal aim of the talk is to give an idea of the general theory of elliptic operators on manifolds with conical singularities and some aspects of their spectral theory, specifically in connection with the existence of rays of minimal growth. I'll begin with a brief review of the notion of manifold with conical singularities, then define what is meant by a differential operator on such manifolds. The analysis of such operators is carried out on the manifold with boundary resulting from blowing up the singularities. After introducing the notion of ellipticity I'll discuss Fredholm properties of closed extensions elliptic operators and end with a theorem on existence of rays of minimal growth obtained in collaboration with J. Gil and T. Krainer. Along the way I'll explain how the theory differs form, but relies on, the theory of elliptic totally characteristic operators.
Peter Hislop
November 18 No seminar No seminar
November 25 Postponed
December 2 Professor Peter Perry, University of Kentucky
TITLE: Solving the mKdV equation with singular initial data by inverse scattering
ABSTRACT : The modified Korteweg-de Vries (mKdV) equation is a completely integrable, nonlinear dispersive equation that can be solved for smooth rapidly decaying initial data by the inverse scattering method. On the other hand, Kato proved long ago that the mKdV equation has weak solutions for initial data in $L^2$, and, most recently, Colliander, Keel, Staffilani, Takaoka, and Tao, proved global well-posedness for initial data in $H^{\alpha}$ if $\alpha>1/4$. In this talk I'll review the inverse scattering method and show how it can be extended to solve the mKdV equation for data in $L^1 \cap L^2$. This work is joint with Rostyslav Hryniv (Lviv) and builds on results of Chris Frayer's thesis.
Changyou Wang
December 9 Professor Jim Brennan, University of Kentucky
TITLE: On a problem of Beurling
ABSTRACT : Let $\Omega$ be an arbitrary bounded simply connected domain in the complex plane, and let w be a positive continuous function on $\Omega$. Denote by $C_w(\Omega)$ the Banach space of all complex-valued functions f for which the product f(z)w(z) is continuous on the closure of $\Omega$ and vanishes on $\partial\Omega$, the norm being defined by $$||f|| = sup_\Omega|f|w.$$ Evidently, the collection of functions Aw(\Omega) = { f \in Cw(\Omega) : f is analytic in \Omega} is a closed subspace of Cw(\Omega). The problem raised by Beurling is to determine whether or not the polynomials are dense in Aw(\Omega). It is my intention to present a solution to this problem which is analogous to Mergeljan’s solution to the classical Bernstein problem for weighted polynomial approximation on the real line, and which applies to the most general regions. In the process I will indicate the manner in which these ideas can be used to strengthen an earlier theorem of Beurling which can be found in his collected works. I will only indicate briefly some of the more technical results that are needed, and which include: 1. a strengthened version of Thomson’s theorem on mean-square polynomial approximation. 2. Tolsa’s theorem on the semi-additivity of analytic capacity. 3. Beurling’s results on quasianalyticity and general distributions.
Changyou Wang