Sharp remainder estimates in the Weyl formula for the Neumann Laplacian on a class of planar regions

We study asymptotic distribution of eigenvalues of the Laplacian on a bounded domain in $ \R^n$. Our main results include an explicit remainder estimate in the Weyl formula for the Dirichlet Laplacian on an arbitrary bounded domain, sufficient conditions for the validity of the Weyl formula for the Neumann Laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula. In particular, we show that the Weyl formula holds true for the Neumann Laplacian on a $ \Lip_\alpha$-domain whenever $ (d-1)/\alpha<d $, prove that the remainder in this formula is $ O(\lambda^{(d-1)/\alpha}) $ and give an example where the remainder estimate $ O(\lambda^{(d-1)/\alpha}) $ is order sharp. We use a new version of variational technique which does not require the extension theorem.

Weil asymptotic formula for the Laplacian on domains with rough boundaries

We study asymptotic distribution of eigenvalues of the Laplacian on a bounded domain in ℝ
 n
 . Our main results include an explicit remainder estimate in the Weyl formula for the Dirichlet Laplacian on an arbitrary bounded domain, sufficient conditions for the validity of the Weyl formula for the Neumann Laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula. In particular, we show that the Weyl formula holds true for the Neumann Laplacian on a Lip
 α
 -domain whenever (d−1)/α<d, prove that the remainder in this formula is O(λ
 (d
 −1)/
 α
 ) and give an example where the remainder estimate O(λ
 (d
 −1)/
 α
 ) is order sharp. We use a new version of the variational technique which does not require the extension theorem.

Weyl Asymptotic Formula for the Laplacian on Domains with Rough Boundaries

We begin by studying semigroup estimates that are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for showing such log--Sobolev inequalities hold for Euclidean regions that satisfy a certain Hardy-type inequality. Our main application is to show that domains with exterior exponential cusps, and hence no Sobolev embedding theorem, satisfy such heat kernel bounds provided the cusp is not too sharp. Finally we consider a rotationally invariant domain with an exponentially sharp cusp and show that ultracontractivity does break down after a certain point.

Log--Sobolev Inequalities and Regions with Exterior Exponential Cusps

Log-Sobolev inequalities and regions with exterior exponential cusps

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Log--Sobolev Inequalities and Regions with Exterior Exponential Cusps

Log-Sobolev inequalities and regions with exterior exponential cusps