# Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

@article{Frank2019TwotermSA,
title={Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain},
author={Rupert L. Frank and Simon Larson},
journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
year={2019},
volume={2020},
pages={195 - 228}
}
• Published 28 January 2019
• Mathematics, Physics
• Journal für die reine und angewandte Mathematik (Crelles Journal)
Abstract We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.

## Figures from this paper

Complementary Asymptotically Sharp Estimates for Eigenvalue Means of Laplacians
• Mathematics
International Mathematics Research Notices
• 2019
We present asymptotically sharp inequalities, containing a 2nd term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin–Li–Yau
Two Consequences of Davies’ Hardy Inequality
• Mathematics
Functional Analysis and Its Applications
• 2021
Abstract Davies’ version of the Hardy inequality gives a lower bound for the Dirichlet integral of a function vanishing on the boundary of a domain in terms of the integral of the squared function
Semiclassical bounds for spectra of biharmonic operators.
• Mathematics
• 2019
The averaged variational principle (AVP) is applied to various biharmonic operators. For the Riesz mean $R_1(z)$ of the eigenvalues we improve the known sharp semiclassical bounds in terms of the
SP ] 2 6 A pr 2 01 9 SEMICLASSICAL BOUNDS FOR SPECTRA OF BIHARMONIC OPERATORS
• 2019
The averaged variational principle (AVP) is applied to various biharmonic operators. For the Riesz mean R1(z) of the eigenvalues we improve the known sharp semiclassical bounds in terms of the volume
Semiclassical asymptotics for a class of singular Schrödinger operators
• Mathematics, Physics
• 2020
Let $\Omega \subset \mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schrodinger operators $-\Delta+ W$ on $\Omega$ with $W(x)\approx\mathrm{dist}(x, \partial\Omega)^{-2}$ as
The Lieb-Thirring inequalities: Recent results and open problems
This review celebrates the generous gift by Ronald and Maxine Linde for the remodeling of the Caltech mathematics department and the author is very grateful to the editors of this volume for the
Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains
For $\Omega \subset \mathbb{R}^n$, a convex and bounded domain, we study the spectrum of $-\Delta_\Omega$ the Dirichlet Laplacian on $\Omega$. For $\Lambda\geq0$ and $\gamma \geq 0$ let
On the spectral asymptotics for the buckling problem
• Mathematics
Journal of Mathematical Physics
• 2021
Abstract. We provide a direct proof of Weyl’s law for the buckling eigenvalues of the biharmonic operator on a wide class of domains of R including bounded Lipschitz domains. The proof relies on
On the error in the two-term Weyl formula for the Dirichlet Laplacian
• Mathematics, Physics
• 2020
We study the optimality of the remainder term in the two-term Weyl law for the Dirichlet Laplacian within the class of Lipschitz regular subsets of $\mathbb{R}^d$. In particular, for the short-time

## References

SHOWING 1-10 OF 45 REFERENCES
Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
• Mathematics, Physics
• 2011
Let −Δ denote the Dirichlet Laplace operator on a bounded open set in Rd. We study the sum of the negative eigenvalues of the operator −h^2Δ − 1 in the semiclassical limit h → 0+. We give a new proof
Semi-classical analysis of the Laplace operator with Robin boundary conditions
• Mathematics
• 2012
We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in
Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator
• Mathematics, Physics
• 2011
We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for
On the asymptotics of the heat equation and bounds on traces associated with the Dirichlet Laplacian
We use the Feynman-Kac formula and a decomposition of the Brownian bridge to obtain pointwise estimates on the diagonal elements of the heat kernel. We also compute bounds on trace (etΔD) where ΔD is
Complementary Asymptotically Sharp Estimates for Eigenvalue Means of Laplacians
• Mathematics
International Mathematics Research Notices
• 2019
We present asymptotically sharp inequalities, containing a 2nd term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin–Li–Yau
AN ESTIMATE NEAR THE BOUNDARY FOR THE SPECTRAL FUNCTION OF THE LAPLACE OPERATOR
The main result of this paper is an estimate as in the title, near a sufficiently smooth part of the boundary of a compact n-dimensional Riemannian manifold Q, for either Dirichlet or Neumann
Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian
• Mathematics
Journal of Spectral Theory
• 2018
We present asymptotically sharp inequalities for the eigenvalues $\mu_k$ of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in
ON THE EIGENVALUES OF THE FIRST BOUNDARY VALUE PROBLEM IN UNBOUNDED DOMAINS
This paper is devoted to the investigation of the spectrum of a polyharmonic operator in unbounded domains. The class of domains for which the spectrum of the corresponding first boundary value
Universal bounds for traces of the Dirichlet Laplace operator
• Physics, Mathematics
• 2010
We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set �¶ �¼ Rd, with d  2. In domains of finite volume the result improves an inequality of
On the Traces of symmetric stable processes on Lipschitz domains
• Mathematics
• 2009
It is shown that the second term in the asymptotic expansion as $t\to 0$ of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order $\alpha$, for any