Let $M$ be a compact Riemannian manifold (without boundary) then the problem $$ \Delta u = - \lambda u; $$ has countably many solutions, called eigenfunctions of $\Delta$, and $\lambda$ are the related eigenvalues; each of them satisfying $$ \| u_j \|_{L^2(M)} =1. $$ Let $\{u_j\}_{j \geq 0}$ be the family of eigenfunctions, ordering in way that the following holds $$ 0=\lambda_0 \leq \lambda_1\leq \ldots\leq\lambda_j \leq \ldots $$ The question is now, is there is an estimate of the form: $$ \sup_{j\geq 0} \| u_j \|_{L^{\infty}(M)} < \infty $$ In the case $M = \mathbb{T}^n = \mathbb{R}^n \,/\,\mathbb{Z}^n$, the $n$-dimensional torus, it is clear because the eigenfunctions can be computed explicity, but in the general case I encountered some problems in using the standard Sobolev embedding theorem.
2026-03-29 19:53:47.1774814027
Uniform estimate for Laplacian eigenfunctions
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