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Katie Gittins

Publications and source records attributed to Katie Gittins.

At least 19 recordsLinked to original sources
arXiv2026

Upper bounds for Courant-sharp Neumann and Robin eigenvalues

We consider the eigenvalues of the Laplacian on an open, bounded, connected set in $\mathbb{R}^n$ with $C^2$ boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a corresponding eigenfunction which achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set. Corrigendum. A previous version of this work was accepted and published by the "Bulletin de la Société Mathématique de France" (see [2] in the bibliography of Appendix B). It contained a gap: the classical (Euclidean) Faber-Krahn inequality was applied in a setting where it might not hold. This version reproduces the previous one with the addition of a corrigendum in Appendix B that addresses the issue. All the results in Sections 2--8 and most of those in Section 9 are thus preserved.

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arXiv2025

Upper bounds for the Steklov eigenvalues of warped products

We obtain upper bounds for the Steklov eigenvalues of warped products $Ω\times_hΣ$, where $Ω$ is a compact Riemannian manifold with boundary and $Σ$ is a closed Riemannian manifold. These bounds involve the volume of $Ω$ and of $\partialΩ$ as well as the eigenvalues of the Laplace operator on the fiber $Σ$ and the $L^p$-norm of the warping function $h$. The bounds are very different depending on the dimension $n$ of the fiber $Σ$ and the value of $p$. In some cases, we obtain optimal upper bounds and stability estimates.

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arXiv2025

Magnetic Steklov operator on differential forms

In this paper, we introduce the magnetic Steklov operator on differential forms and show that the underlying boundary value problem is well-posed. Moreover, we show that an analogue of the Diamagnetic Inequality does not always hold for this operator, and we present some spectral computations of magnetic Steklov operators for $2$-dimensional and $4$-dimensional balls in Euclidean space.

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arXiv2025

Qualitative properties of the heat content

We obtain monotonicity and convexity results for the heat content of domains in Riemannian manifolds and in Euclidean space subject to various initial temperature conditions. We introduce the notion of a strictly decreasing temperature set, and show that it is a sufficient condition to ensure monotone heat content. In addition, in Euclidean space, we construct a domain and an initial condition for which the heat content is not monotone, as well as a domain and an initial condition for which the heat content is monotone but not convex.

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arXiv2025

Nodal counts for the Robin problem on Lipschitz domains

We consider the Courant-sharp eigenvalues of the Robin Laplacian for bounded, connected, open sets in $\mathbb{R}^n$, $n \geq 2$, with Lipschitz boundary. We prove Pleijel's theorem which implies that there are only finitely many Courant-sharp eigenvalues in this setting as well as an improved version of Pleijel's theorem, extending previously known results that required more regularity of the boundary. In addition, we obtain an upper bound for the number of Courant-sharp Robin eigenvalues of a bounded, connected, convex, open set in $\mathbb{R}^n$ with $C^2$ boundary that is explicit in terms of the geometric quantities of the set and the norm sup of the negative part of the Robin parameter.

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arXiv2025

A note on the magnetic Steklov operator on functions

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger-Jammes type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias and Savo. In addition, we compute the full spectrum in the case of the Euclidean $2$-ball and $4$-ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the non-magnetic case due to Colbois, Girouard and Hassannezhad.

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arXiv2024

Spectral ratios and gaps for Steklov eigenvalues of balls with revolution-type metrics

We investigate upper bounds for the spectral ratios and gaps for the Steklov eigenvalues of balls with revolution-type metrics. We do not impose conditions on the Ricci curvature or on the convexity of the boundary. We obtain optimal upper bounds for the Steklov spectral ratios in dimensions 3 and higher. In dimension 3, we also obtain optimal upper bounds for the Steklov spectral gaps. By imposing additional constraints on the metric, we obtain upper bounds for the Steklov spectral gaps in dimensions 4 and higher.

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arXiv2024

Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms

In this paper we introduce the magnetic Hodge Laplacian, which is a generalization of the magnetic Laplacian on functions to differential forms. We consider various spectral results, which are known for the magnetic Laplacian on functions or for the Hodge Laplacian on differential forms, and discuss similarities and differences of this new ``magnetic-type'' operator.

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arXiv2024

Do the Hodge spectra distinguish orbifolds from manifolds? Part 2

In \cite{GGKM-SSS} we examined the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We showed that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds from manifolds as long as the singular sets have codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le 3.$ Here we give both positive and negative inverse spectral results for the individual $p$-spectra considered separately. For example, we give conditions on the codimension of the singular set which guarantee that the volume of the singular set is determined, and in many cases we show by providing counterexamples that the conditions are sharp.

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arXiv2022

Heat flow in polygons with reflecting edges

We investigate the heat flow in an open, bounded set $D$ in $\mathbb{R}^2$ with polygonal boundary $\partial D$. We suppose that $D$ contains an open, bounded set $\widetilde{D}$ with polygonal boundary $\partial \widetilde{D}$. The initial condition is the indicator function of $\widetilde{D}$ and we impose a Neumann boundary condition on the edges of $\partial D$. We obtain an asymptotic formula for the heat content of $\widetilde{D}$ in $D$ as time $t\downarrow 0$.

math.AP↗
arXiv2021

Do the Hodge spectra distinguish orbifolds from manifolds? Part 1

We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds with singularities from manifolds as long as the singular sets have codimension $\le 3.$ This is enough to distinguish orbifolds from manifolds for dimension $\le 3.$

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arXiv2020

Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index

We obtain upper bounds for the Steklov eigenvalues $σ_k(M)$ of a smooth, compact, connected, $n$-dimensional submanifold $M$ of Euclidean space with boundary $Σ$ that involve the intersection indices of $M$ and of $Σ$. One of our main results is an explicit upper bound in terms of the intersection index of $Σ$, the volume of $Σ$ and the volume of $M$ as well as dimensional constants. By also taking the injectivity radius of $Σ$ into account, we obtain an upper bound that has the optimal exponent of $k$ with respect to the asymptotics of the Steklov eigenvalues as $k \to \infty$.

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arXiv2019

Heat flow from polygons

We study the heat flow from an open, bounded set $D$ in $\R^2$ with a polygonal boundary $\partial D$. The initial condition is the indicator function of $D$. A Dirichlet $0$ boundary condition has been imposed on some but not all of the edges of $\partial D$. We calculate the heat content of $D$ in $\R^2$ at $t$ up to an exponentially small remainder as $t\downarrow 0$.

math.AP↗
arXiv2019

Courant-sharp Robin eigenvalues for the square -- the case with small Robin parameter--

This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter $h$). For the square, our first paper focused on the case where $h$ is large and extended results that were obtained by Pleijel, Bérard--Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of $h$, where $h$ is positive and not close to $0$. In this second paper, we extend results that were obtained by Helffer--Persson-Sundqvist for the Neumann problem to the case where $h>0$ is small.

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arXiv2019

Courant-sharp Robin eigenvalues for the square and other planar domains

This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, Bérard--Helffer, Helffer--Persson--Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter $h$, we focus on the case when $h$ is large. We hope to come back to the analysis when $h$ is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with $C^{2,α}$ boundary ($α>0$) as $h$ large varies.

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arXiv2018

Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is isospectral to the disk.

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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.

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