Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary

Details Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary

2001-03-13

arxiv.org/abs/math/0103080v2

Mathematics

Spectral Theory

12 pages, correction of minor errors in Sections 4, 5; to appear in Comm. PDE

Scientific paper

Let $u$ be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let $-\lambda^2$ be the corresponding eigenvalue. We consider the problem of estimating the maximum of $u$ in terms of $\lambda$, for large $\lambda$, assuming $u$ is $L^2$-normalized. We prove that $\max_M u\leq C_M \lambda^{(n-1)/2}$, which is optimal for some $M$. Our proof simplifies some of the arguments used before for such problems. In order to make the article accessible to non-specialists, we review the 'wave equation method' (which has become standard in asymptotic eigenvalue problems) and discuss some special cases which may be handled by more direct methods.

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