UNIFORM BOUNDS FOR EIGENFUNCTIONS OF THE LAPLACIAN ON MANIFOLDS WITH BOUNDARY*

By D. Grieser. 2001

112 citations recorded by the source corpus.

ABSTRACT Let u be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let −λ2 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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d7266a3c2377c9aae5e85ac2cd3cd80311708d27

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Citations in the corpus

Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations (2005, background): “We note that in general {Cn} is unbounded [30].”
On the Lp norm of spectral clusters for compact manifolds with boundary (2006, methodology): “[5] Grieser, D., Lp bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries.”
Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law (2014, methodology): “For L2-normalized eigenfunctions and quasi-modes, upper bounds of the form Cλn+1 for the L∞ norms of the gradient are well-known and extensively used (cf [12, 13, 33, 34, 4, 5, 6, 38]).”