# Spectral inequalities in quantitative form

@inproceedings{Brasco2016SpectralII, title={Spectral inequalities in quantitative form}, author={Lorenzo Brasco and Guido De Philippis}, year={2016} }

We review some results about quantitative improvements of sharp inequalities for eigenvalues of the Laplacian.

## 27 Citations

### A quantitative Bucur–Henrot inequality

- Mathematics
- 2022

In this paper, we prove a quantitative version of the isoperimetric inequality involving the second non‐trivial eigenvalue of the Laplacian with Neumann boundary condition established by Bucur and…

### A quantitative Weinstock inequality

- Mathematics
- 2019

The paper is devoted to the study of a quantitative Weinstock inequality in higher dimension for the first non trivial Steklov eigenvalue of Laplace operator for convex sets. The key rule is played…

### On a conjecture of Ashbaugh and Benguria about lower eigenvalues of the Neumann laplacian

- MathematicsMathematische Annalen
- 2022

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains in a Euclidean space or a hyperbolic space which strengthens the…

### BEST CONSTANTS IN SOME ESTIMATES FOR THE HARMONIC MAXIMAL OPERATOR ON THE REAL LINE

- Mathematics
- 2020

The paper contains the proofs of strong-type, weaktype, Lorentz-norm and stability estimates for the harmonic maximal operator on the real line, associated with an arbitrary Borel measure. The…

### The sharp quantitative isocapacitary inequality

- MathematicsRevista Matemática Iberoamericana
- 2021

We prove a sharp quantitative form of the classical isocapacitary inequality. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the…

### BEST CONSTANTS IN SOME ESTIMATES FOR THE HARMONIC MAXIMAL OPERATOR ON THE REAL LINE

- Mathematics
- 2020

. The paper contains the proofs of strong-type, weak-type, Lorentz-norm and stability estimates for the harmonic maximal operator on the real line, associated with an arbitrary Borel measure. The…

### Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants

- Mathematics, Computer Science
- 2020

We provide a quantitative lower bound to the Cheeger constant of a set $\Omega$ in both the Euclidean and the Gaussian settings in terms of suitable asymmetry indexes. We provide examples which show…

### Well-Posedness of Weinberger’s Center of Mass by Euclidean Energy Minimization

- Mathematics
- 2020

The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure.

## References

SHOWING 1-10 OF 75 REFERENCES

### The sharp quantitative isoperimetric inequality

- Mathematics
- 2008

A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall.

### A review of Hardy inequalities

- Mathematics
- 1998

We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to…

### The quantitative isoperimetric inequality and related topics

- Mathematics
- 2015

We present some recent stability results concerning the isoperimetric inequality and other related geometric and functional inequalities. The main techniques and approaches to this field are…

### Isoperimetric Inequalities for Eigenvalues of the Laplacian

- Mathematics
- 2007

These are extended notes based on the series of four lectures on isoperimetric inequalities for the Laplacian, given by the author at the Arizona School of Analysis with Applications, in March 2010.

### Isoperimetric inequalities in mathematical physics

- Physics, Mathematics
- 1951

The description for this book, Isoperimetric Inequalities in Mathematical Physics. (AM-27), will be forthcoming.

### Best constant in Sobolev inequality

- Mathematics, Materials Science
- 1976

SummaryThe best constant for the simplest Sobolev inequality is exhibited. The proof is accomplished by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus…

### Sharp Stability of Some Spectral Inequalities

- Mathematics
- 2012

In this work we review two classical isoperimetric inequalities involving eigenvalues of the Laplacian, both with Dirichlet and Neumann boundary conditions. The first one is classically attributed to…

### A quantitative Pólya-Szegö principle

- Mathematics
- 2008

Abstract The radially decreasing symmetrization is well known not to increase Dirichlet type integrals of Sobolev functions. In the present paper, the deviation of a function from its symmetral is…

### Sharp stability inequalities for the Plateau problem

- Mathematics
- 2014

The validity of global quadratic stability inequalities for uniquely regular area minimizing hypersurfaces is proved to be equivalent to the uniform positivity of the second variation of the area.…