UNIFORM BOUNDS FOR EIGENFUNCTIONS OF THE LAPLACIAN ON MANIFOLDS WITH BOUNDARY

Let u be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let −λ 2 be the corresponding eigenvalue. We consider the problem of estimating max M u in terms of λ, for large λ, assuming ∫ M u 2 = 1. We prove that max M u ≤C M λ (n − 1)/2 , which is optimal for some M. Our proof simplifies some of the arguments used before for such problems. We review the 'wave equation method' and discuss some special cases which may be handled by more direct methods. *This work was partially supported by NSF grant DMS-9306389 and by the Mathematical Sciences Research Institute, Berkeley.