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Seminar

Language	English
From	2011/12/16 15:00
To	2011/12/16 16:00
Place	Room 251, Building No.3, Faculty of Science, Kyoto University
Seminar Name	NLPDE Seminar
Title	On the fractal dimension of divergence sets for Sch $\ddot{o}$ dinger equations
Field	Analysis
Speakers	Keith Rogers
Affiliation	Instituto de Ciencias Mathematicas
Abstract	We will consider the Schr $\ddot{o}$ dinger equation, $i\partial_tu+\Delta u=0$ , in $\mathbb{R}^{1+1}$ , with initial data $u_0$ in potential spaces $H^s(\mathbb{R})$ . Carleson proved that the solution converges, almost everywhere with respect to Lebesgue measure, to $u_0$ along the straight lines $t\to (x,t)$ as $t\to 0$ when $s\ge 1/4$ . We improve this result in two ways. Firstly we show that the convergence holds everywhere apart from a set of Hausdorff dimension less than or equal to $1-2s$ when $1/4\le s\le 1/2$ , and that this is sharp. Secondly we will prove that the convergence holds when the straight lines are replaced by continuously differentiable curves. This allows us to refine results of Sj $\ddot{o}$ gren--Torrea and Yajima for the quantum harmonic oscillator. This is joint work with J.A. Barcel $\dot{o}$ , J. Bennett, A. Carbery and S. Lee.
Link	http://www.math.kyoto-u.ac.jp/~yosihiro/nlpde/nlpde.html