Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains

@article{Larson2019AsymptoticSO,
  title={Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains},
  author={Simon Larson},
  journal={Journal of Spectral Theory},
  year={2019}
}
  • Simon Larson
  • Published 17 November 2016
  • Mathematics
  • Journal of Spectral Theory
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 $\Omega_{\Lambda, \gamma}(\mathcal{A})$ denote any extremal set of the shape optimization problem $$ \sup\{ \mathrm{Tr}(-\Delta_\Omega-\Lambda)_-^\gamma: \Omega \in \mathcal{A}, |\Omega|=1\}, $$ where $\mathcal{A}$ is an admissible family of convex domains in $\mathbb{R}^n$. If $\gamma \geq 1$ and $\{\Lambda_j… 
Improved Bounds for Hermite–Hadamard Inequalities in Higher Dimensions
Let $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n be a convex domain and let $$f:\Omega \rightarrow {\mathbb {R}}$$ f : Ω → R be a positive, subharmonic function (i.e., $$\Delta f \ge 0$$ Δ f ≥ 0 ).
Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues
We prove that in dimension $$n \ge 2$$n≥2, within the collection of unit-measure cuboids in $$\mathbb {R}^n$$Rn (i.e. domains of the form $$\prod _{i=1}^{n}(0, a_n)$$∏i=1n(0,an)), any sequence of
Optimal unions of scaled copies of domains and Pólya's conjecture
Given a bounded Euclidean domain $\Omega$, we consider the sequence of optimisers of the $k^{\rm th}$ Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled
Optimal stretching for lattice points under convex curves
Suppose we count the positive integer lattice points beneath a convex decreasing curve in the first quadrant having equal intercepts. Then stretch in the coordinate directions so as to preserve the
Shifted lattices and asymptotically optimal ellipses
Translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions, and consider a decreasing concave (or convex) curve in the first
Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles
We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a
Asymptotic Behaviour of Extremal Averages of Laplacian Eigenvalues
We study the convergence of extrema of averages of eigenvalues of the Dirichlet Laplacian on domains in $$\mathbb {R}^{n}$$Rn under both measure and surface measure restrictions. In the former case
Two Consequences of Davies’ Hardy Inequality
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
Optimal stretching for lattice points and eigenvalues
We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically
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
...
1
2
...

References

SHOWING 1-10 OF 59 REFERENCES
On the minimization of Dirichlet eigenvalues
Results are obtained for two minimization problems: $$I_k(c)=\inf \{\lambda_k(\Omega): \Omega\ \textup{open, convex in}\ \mathbb{R}^m,\ \mathcal{T}(\Omega)= c \},$$ and
Lipschitz Regularity of the Eigenfunctions on Optimal Domains
AbstractWe study the optimal sets $${\Omega^\ast\subseteq\mathbb{R}^d}$$Ω*⊆Rd for spectral functionals of the form $${F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}$$F(λ1(Ω),…,λp(Ω)), which
Regularity of the optimal sets for some spectral functionals
AbstractIn this paper we study the regularity of the optimal sets for the shape optimization problem $$\min\Big\{\lambda_{1}(\Omega)+\dots+\lambda_{k}(\Omega) : \Omega \subset {\mathbb {R}}^{d} {\rm
Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint
In this paper we prove that the shape optimization problem $$\min \bigl\{\lambda_k(\varOmega):\ \varOmega\subset \mathbb{R}^d,\ \varOmega\ \hbox{open},\ P(\varOmega)=1,\ |\varOmega|<+\infty \bigr\},
Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues
We prove that in dimension $$n \ge 2$$n≥2, within the collection of unit-measure cuboids in $$\mathbb {R}^n$$Rn (i.e. domains of the form $$\prod _{i=1}^{n}(0, a_n)$$∏i=1n(0,an)), any sequence of
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
On the Schrödinger equation and the eigenvalue problem
AbstractIf λk is thekth eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝn, H. Weyl's asymptotic formula asserts that $$\lambda _k \sim C_n \left( {\frac{k}{{V(D)}}}
On the Minimization of Dirichlet Eigenvalues of the Laplace Operator
We study variational problems of the form $$\inf\{\lambda_k(\Omega): \Omega\ \mbox{open in}\ \mathbb{R}^m,\ T(\Omega ) \le1 \},$$ where λk(Ω) is the k-th eigenvalue of the Dirichlet Laplacian acting
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
Recent Existence Results for Spectral Problems
In this survey we present the new techniques developed for proving existence of optimal sets when one minimizes functionals depending on the eigenvalues of the Dirichlet Laplacian with a measure
...
1
2
3
4
5
...