PDE Seminar Abstracts

Xiaolong Han

Australian National University, Australia

Tuesday 21 April 2015, 12-1pm, Carslaw Room 829 (AGR)

Australian National University, Australia

Tuesday 21 April 2015, 12-1pm, Carslaw Room 829 (AGR)

Sogges ${L}^{p}$-estimates give upper bounds to the ${L}^{p}$-norms of ${L}^{2}$-normalized eigenfunctions on smooth and compact manifolds. They are also sharp on the sphere, where Gaussian beams saturate the maximal ${L}^{p}$-norm growth for small $p$ and zonal harmonics for large $p$. But these maximisers are very sparse in the orthonormal basis of eigenfunctions (ONBE); their densities are both zero. Sogge and Zelditch proposed the question of whether there exists a manifold supporting an ONBE that contains a positive density subsequence of maximisers. In this talk, I will give the positive answer to this question and construct such example for all small $p$.

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