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.
Uniform estimate for Laplacian eigenfunctions
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pde
lp-spaces
laplacian
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1I googled 'bound eigenfunctions laplacian" and got [this](http://mathoverflow.net/questions/118692/uniform-bound-on-the-eigenfunctions-of-the-laplacian) as second result (your question being first) – 2017-02-07
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0Thank you for your advice, but the question is if you can built an estimate which does not depends on the eigenvalues. Remember that the eingenvalues of the Laplacian form an increasing sequence which goes to infinity, And for this reason, the inequality in the answer that you linked me, does not answer to my question. – 2017-02-07