FALL 2021 Nonlinear Analysis Seminar Series

Professor Sagun Chanillo, Rutgers University

Title: Local Version of Courant's Nodal domain theorem.

Abstract:

Consider a smooth, compact Riemannian manifold with no boundary, endowed with a smooth metric. A famous theorem of Courant states that the k-th eigenfunction for the Laplace-Beltrami operator can have at most k nodal domains. Nodal domains are the open and connected sets where the eigenfunction does not vanish. H. Donnelly and Fefferman obtained some 30 years ago a local version of this theorem. Improvements were made by Chanillo-Muckenhoupt and others. In this talk we obtain the optimal local version of the local Courant theorem. We also relate this result to conjectures of S.-T. Yau on nodal sets, that is the zero set of eigenfunctions. The results of our talk have been obtained jointly with A. Logunov, E. Mallinikova and D. Mangoubi.

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