| Analysis Research Group | |
| Analysis
Seminar (Fall 2008) Wednesday 4:30 - 5:20 pm Cardwell 120 (except when otherwise noted) |
Organizer: Charles Moore Previous semesters: Spring 06 Fall 06 Spring 07 Fall 07 Spring 08 |
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Wednesday, August 27- Peiyong
Wang, Wayne State University A formula for p-harmonic functions
Wednesday, September 10- Petra Bonfert-Taylor, Wesleyan University Distortion Questions for Discrete Quasiconformal Groups A discrete quasiconformal group is a discrete group of quasiconformal homeomorphisms of $\overline{\mathbb{R}}^n$ having a uniform (upper) bound on the dilatation. Somewhat surprisingly, there is a good theory for these groups involving an exponent of convergence, a Patterson-Sullivan measure,etc., and the limit set of the group. As to be expected from the loss of a conformal action, the theory, as initiated by Patterson, Sullivan, and Tukia, is considerably more complicated than the usual theory applied to discrete Möbius groups. For a non-elementary discrete group of Möbius transformations it was shown that the exponent of convergence equals the Hausdorff dimension of the conical limit set of the group (Bishop & Jones, Patterson, Sullivan, and Tukia). In this talk we will discuss a generalization of this result to discrete quasiconformal groups: The exponent of convergence is an upper bound on the Hausdorff dimension of the conical limit set, with strict inequality possible. For planar quasiconformal groups, using Astala's results on the integrability of the Jacobian of a quasiconformal mapping,we establish sharp bounds on these two quantities in terms of each other and the dilatation of the group. Monday, September 22- Gustavo Ponce, University of California, Santa Barbara Decay estimates and unique continuation properties of solutions to linear and non-linear Schrödinger equations (joint works with L. Escauriaza, C. E. Kenig, and L. Vega, and J. Nahas.) We shall present some results concerning decay properties of solutions of the Schrödinger equations. The first of them is related with estimates for the Leibnitz rule for fractional derivatives. The second one is concerned with exponential decay solutions for the Schrödinger equation. For the case of the free Schrödinger equation we deduce a convexity estimate for Gaussian weight, and the corresponding version of the Hardy's uncertainty principle. We present extensions of these results to the case of Schrödinger equation with potential (in both cases, with potential independent and depending on time). Finally, we apply some of these results to establish some unique continuation results for the semi-linear Schrödinger equation. Wednesday, October 1 - Chuck Moore Uniqueness for the two-phase Stefan problem I will show that solutions to the two-phase Stefan problem are unique, that is, if two solutions agree initially in the sense of measures, then they agree a.e. for t>0. Wednesday, October 8- Chuck Moore Uniqueness for the two-phase Stefan problem, continued Wednesday, October 15 - Two short talks by Maria J. Carro and Javier Soria, University of Barcelona, visiting University of Kansas Maria J. Carro: End-point estimates for several kind of operators. The purpose of this talk is to present 3 different problems in Harmonic Analysis that can be studied using the so-called Yano's extrapolation theory. This theory started in 1951 when Yano proved that if a sublineraroperator is bounded from L^p into L^p with constant 1/(p-1) then it is bounded in the Orlicz space L \log L. In our days, the theory is still of great interest since there are many open questions to be solved. Javier Soria: When is a quasi-norm equivalent to a norm? Applications to weak-type spaces. We will review a necessary and sufficient condition for a quasi-norm to be normable, which leads us to consider the decomposition norm. For particular examples (weak-type and other Banach function spaces) we will find the best constants for the equivalence of norms, showing also the optimal triangle inequality. Wednesday, October 22 - Virginia Naibo Boundedness of bilinear operators on products of Besov and Lebesgue We discuss Besov-Lebesgue mapping properties for families of bilinear operators, including molecular paraproducts and Hörmander-Mihlin multipliers, and their connections with bilinear Littlewood-Paley theory. This is joint work with Diego Maldonado. Wednesday, October 29 - Marianne Korten A Fatou Theorem for the one-phase Stefan problem We show the existence of limits through nontangential parabolic cones for positive solutions of the Stefan problem. Wednesday, November 5- Marianne Korten A Fatou Theorem for the one-phase Stefan problem, continued Wednesday, November 19 - Ray Treinen Numerical Simulations of Hele-Shaw and Stefan Problems Two free boundary problems are considered here: Hele-Shaw and Stefan. The symmetric case is explored numerically using a spectral method. In addition to initial data and data given on the slot, the free boundary satisfies a condition that depends on both the normal derivative there, as well as the (mean) curvature. Wednesday, December 3- Nguyen Hoang A discrepancy principle for equations with monotone continuous operators A discrepancy principle for solving nonlinear equztions with monotone operators given noisy data is formulated. The existence and uniqueness of the corresponding regularization parameter a(delta) are proved. Convergence of the solution obtained by the discrepancy principle is justified. The results are obtained under natural assumptions on the nonlinear operator. The talk is based on a joint work with Prof. A. G. Ramm. Wednesday, December 10- Nguyen Hoang An iterative scheme for solving equations with monotone operators An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and posteriori stopping rules for the iterative scheme are formulated and justified. The talk is based on a joint work with Prof. A.G. Ramm. |