# Two Consequences of Davies’ Hardy Inequality

@article{Frank2021TwoCO, title={Two Consequences of Davies’ Hardy Inequality}, author={Rupert L. Frank and Simon Larson}, journal={Functional Analysis and Its Applications}, year={2021}, volume={55}, pages={174-177} }

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 with a weight containing the averaged distance to the boundary. This inequality is applied to easily derive two classical results of spectral theory, E. Lieb’s inequality for the first eigenvalue of the Dirichlet Laplacian and G. Rozenblum’s estimate for the spectral counting function of the Laplacian…

## References

SHOWING 1-9 OF 9 REFERENCES

ON THE EIGENVALUES OF THE FIRST BOUNDARY VALUE PROBLEM IN UNBOUNDED DOMAINS

- Mathematics
- 1972

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…

Hardy-Sobolev-Maz'ya inequalities for arbitrary domains

- Mathematics
- 2011

We prove a Hardy-Sobolev-Maz’ya inequality for arbitrary domains Ω ⊂ R^N with a constant depending only on the dimension N ≥ 3. In particular, for convex domains this settles a conjecture by…

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

- Mathematics, PhysicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2019

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…

On the lowest eigenvalue of the Laplacian for the intersection of two domains

- Mathematics
- 1983

AbstractIfA andB are two bounded domains in ℝn and λ(A), λ(B) are the lowest eigenvalues of −Δ with Dirichlet boundary conditions then there is some translate,Bx, ofB such that λ(A∩Bx)<λ(A)+λ(B). A…

Can One See the Fundamental Frequency of a Drum?

- Mathematics, Physics
- 2005

AbstractWe establish two-sided estimates for the fundamental frequency (the lowest eigenvalue) of the Laplacian in an open set
$$\Omega \subset \mathbb{R}^n$$ with the Dirichlet boundary condition.…

The Lieb-Thirring inequalities: Recent results and open problems

- Physics, Mathematics
- 2020

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

- MathematicsJournal of Spectral Theory
- 2019

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…

the uncertainty principle

- Mathematics
- 1983

On considere l'existence et la regularite des solutions d'equations aux derivees partielles, la construction de solutions fondamentales explicites et les valeurs propres d'operateurs de Schrodinger

Some norm bounds and quadratic form inequalities for Schrödinger operators

- II. J. Operator Theory 12
- 1984