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Simon Larson

Publications and source records attributed to Simon Larson.

29 records · Page 2Linked to original sources
arXiv2020

A sharp multidimensional Hermite-Hadamard inequality

Let $Ω\subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon Ω\to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|Ω|}\int_Ωf(x)\,dx \leq \frac{d}{|\partialΩ|}\int_{\partialΩ} f(x)\,dσ(x)\,. \] Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $Ω\subset \mathbb{R}^d$ is a bounded convex domain and $u$ is the solution of $-Δu =1$ with homogeneous Dirichlet boundary conditions, then \[ \|\nabla u\|_{L^\infty(Ω)} < d\frac{|Ω|}{|\partialΩ|}\,. \] Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al.

math.CA↗
arXiv2020

Lieb-Thirring inequalities for wave functions vanishing on the diagonal set

We propose a general strategy to derive Lieb-Thirring inequalities for scale-covariant quantum many-body systems. As an application, we obtain a generalization of the Lieb-Thirring inequality to wave functions vanishing on the diagonal set of the configuration space, without any statistical assumption on the particles.

math-ph↗
arXiv2020

A bound for the perimeter of inner parallel bodies

We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body $Ω$. The bound depends only on the perimeter and inradius $r$ of the original body and states that \[|\partialΩ_t| \geq \Bigl(1-\frac{t}{r}\Bigr)^{n-1}_+ |\partial Ω|.\] In particular the bound is independent of any regularity properties of $\partialΩ$. As a by-product of the proof we establish precise conditions for equality. The proof, which is straightforward, is based on the construction of an extremal set for a certain optimization problem and the use of basic properties of mixed volumes. This is a revised version of the paper published in J. Funct. Anal. (2016) where an error is addressed in accordance with a corrigendum to appear in J. Funct. Anal. The main result of the paper remains the same but an error in Lemma 2.1 has been corrected and the subsequent proofs have been adapted accordingly.

math.MG↗
arXiv2020

On the error in the two-term Weyl formula for the Dirichlet Laplacian

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 asymptotics of the trace of the heat kernel we prove that the error term cannot be made quantitatively better than little-$o$ of the second term.

math.SP↗
arXiv2019

Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $Δf \geq 0$). Then $$ \frac{1}{|Ω|} \int_Ω{f dx} \leq \frac{c_n}{ |\partial Ω| } \int_{\partial Ω}{ f dσ},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ Ω_2 \subset Ω_1 \subset \mathbb{R}^n$: $$ \frac{|\partial Ω_1|}{|Ω_1|} \frac{| Ω_2|}{|\partial Ω_2|} \leq n.$$

math.CA↗
arXiv2018

Maximizing Riesz means of anisotropic harmonic oscillators

We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of $(\mathbb{N}-\tfrac12)\times(\mathbb{N}-\tfrac12)$ inside triangles with vertices $(0, 0), (0, λ\sqrtβ)$ and $(λ/{\sqrtβ}, 0)$ with respect to $β>0$, for fixed $λ\geq 0$. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice $(\mathbb{N}+σ)\times(\mathbb{N}+τ)$, for $σ, τ>-1$. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, $σ=τ=-\tfrac12$, is a critical case.

math.SP↗
arXiv2017

Asymptotic behaviour of cuboids optimising Laplacian eigenvalues

We prove that in dimension $n \geq 2$, within the collection of unit measure cuboids in $\mathbb{R}^n$ (i.e. domains of the form $\prod_{i=1}^{n}(0, a_n)$), any sequence of minimising domains $R_k^\mathcal{D}$ for the Dirichlet eigenvalues $λ_k$ converges to the unit cube as $k \to \infty$. Correspondingly we also prove that any sequence of maximising domains $R_k^\mathcal{N}$ for the Neumann eigenvalues $μ_k$ within the same collection of domains converges to the unit cube as $k\to \infty$. For $n=2$ this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for $n=3$ was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as $k \to \infty$. We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first $k$ eigenvalues.

math.SP↗
arXiv2017

Exclusion bounds for extended anyons

We introduce a rigorous approach to the many-body spectral theory of extended anyons, i.e. quantum particles confined to two dimensions that interact via attached magnetic fluxes of finite extent. Our main results are many-body magnetic Hardy inequalities and local exclusion principles for these particles, leading to estimates for the ground-state energy of the anyon gas over the full range of the parameters. This brings out further non-trivial aspects in the dependence on the anyonic statistics parameter, and also gives improvements in the ideal (non-extended) case.

math-ph↗
arXiv2016

On the remainder term of the Berezin inequality on a convex domain

We study the Dirichlet eigenvalues of the Laplacian on a convex domain in $\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper bounds for the Riesz means of order $σ\geq 3/2$ established in an article by Geisinger, Laptev and Weidl. This is achieved by refining estimates for a negative second term in the Berezin inequality. The obtained remainder term reflects the correct order of growth in the semi-classical limit and depends only on the measure of the boundary of the domain. We emphasize that such an improvement is for general $Ω\subset\mathbb{R}^n$ not possible and was previously known to hold only for planar convex domains satisfying certain geometric conditions. As a corollary we obtain lower bounds for the individual eigenvalues $λ_k$, which for a certain range of $k$ improves the Li--Yau inequality for convex domains. However, for convex domains one can use different methods to obtain even stronger such lower bounds.

math.SP↗
arXiv2016

Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $Ω$ in the Heisenberg group $\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $Ω$ is the half-space given by $\langle ξ, ν\rangle > d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $Ω$ the inequality is sharp and takes the form \begin{equation} \int_Ω|\nabla_{\mathbb{H}^n}u|^2 \, dξ\geq \frac{1}{4}\int_Ω \sum_{i=1}^n\frac{\langle X_i(ξ), ν\rangle^2+\langle Y_i(ξ), ν\rangle^2}{\textrm{dist}(ξ, \partial Ω)^2}|u|^2\, dξ, \end{equation} where $\textrm{dist}(\, \cdot\,, \partial Ω)$ denotes the Euclidean distance from $\partial Ω$.

math.AP↗