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Mathematics Colloquium


Date:  Tuesday, November 27, 2012

Title:  The Weyl law and bounds for the forced membrane problem

Abstract:  Using a combination of classical ideas, some going back to Sommerfeld a hundred years ago, and more modern techniques, we address a question raised by Dos Santos Ferreira, Kenig and Salo about what are the optimal regions ${\mathcal R}\subset {\mathbb C}$ for which one can have uniform Lebesgue-estimates for solutions of the forced membrane equation, $$(\Delta+\zeta)u=f,$$ as $\zeta$ ranges over ${\mathcal R}$. We confirm, as Sommerfeld reasoned, that properties of the solution $u$ should depend on properties of solutions of the stationary equations $(\Delta+\lambda)u=0$ (the eigenfunctions) and properties of the spectrum (the eigenvalues $-\lambda$ of the Laplacian). This is joint work with J. Bourgain, P. Shao and X. Yao.

Speaker:  Christopher Sogge
Institution:  Johns Hopkins University


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