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Graham Cox

Publications and source records attributed to Graham Cox.

35 records · Page 2Linked to original sources
arXiv2020

Exponential dichotomies for elliptic PDE on radial domains

It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is ill-posed, in the sense that solutions do not typically exist forward or backward in time. In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE. This generalizes the spatial dynamics approach, which applies to infinite cylindrical (channel) domains, and also generalizes previous work on radial domains as we impose no symmetry assumptions on the equation or its solutions.

math.AP↗
arXiv2020

Limiting Eigenfunctions of Sturm-Liouville operators Subject to a Spectral Flow

We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described in a previor result of the last two authors with Berkolaiko, where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization.

math.SP↗
arXiv2020

The ponderomotive Lorentz force

This paper describes a curious phenomenon: a particle in a rapidly varying potential is subject to an effective magnetic-like force. This force is in addition to the well-known ponderomotive force, but it has not been shown to exist before except for the linear case of a rapidly rotating quadratic saddle potential. We show that this is a universal phenomenon: the magnetic-like force arises generically in potential force fields with rapid periodic time dependence, including but not limited to rotational dependence.

math.DS↗
arXiv2019

Isoperimetric relations between Dirichlet and Neumann eigenvalues

Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture that this number is controlled by the isoperimetric ratio of the domain. This has applications to the nodal deficiency of eigenfunctions and is closely related to a long-standing conjecture of Yau on the Hausdorff measure of nodal sets.

math.AP↗
arXiv2019

Conformal invariants from nodal sets II. Manifolds with boundary

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators on manifolds with boundary. We also consider applications to curvature prescription problems on manifolds with boundary. We relate Dirichlet and Neumann eigenvalues and put the results developed here for the Escobar problem into the more general framework of boundary operators of arbitrary order.

math.AP↗
arXiv2017

A symplectic perspective on constrained eigenvalue problems

The Maslov index is a powerful tool for computing spectra of selfadjoint, elliptic boundary value problems. This is done by counting intersections of a fixed Lagrangian subspace, which designates the boundary condition, with the set of Cauchy data for the differential operator. We apply this methodology to constrained eigenvalue problems, in which the operator is restricted to a (not necessarily invariant) subspace. The Maslov index is defined and used to compute the Morse index of the constrained operator. We then prove a constrained Morse index theorem, which says that the Morse index of the constrained problem equals the number of constrained conjugate points, counted with multiplicity, and give an application to the nonlinear Schrödinger equation.

math.SP↗
arXiv2017

Instability of pulses in gradient reaction-diffusion systems: A symplectic approach

In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction-diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evolution equation. Finally, we use a symmetry argument to show that any pulse solution must have nonzero Maslov index, and hence be unstable.

math.DS↗
arXiv2017

Gaussian curvature and gyroscopes

We relate Gaussian curvature to the gyroscopic force, thus giving a mechanical interpretation of the former and a geometrical interpretation of the latter. We do so by considering the motion of a spinning disk constrained to be tangent to a curved surface. It is shown that the spin gives rise to a force on the disk which is equal to the magnetic force on a point charge moving in a magnetic field normal to the surface, of magnitude equal to the Gaussian curvature, and of charge equal to the disk's axial spin. In a special case, this demonstrates that the precession of Lagrange's top is due to the curvature of a sphere determined by the parameters of the top.

math.DS↗
arXiv2016

Manifold decompositions and indices of Schrödinger operators

The Maslov index is used to compute the spectra of different boundary value problems for Schrödinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold $M$ divided into components $Ω_1$ and $Ω_2$ by a separating hypersurface $Σ$. A homotopy argument relates the spectrum of a second-order elliptic operator on $M$ to its Dirichlet and Neumann spectra on $Ω_1$ and $Ω_2$, with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslov index can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on $Σ$. Applications are given to doubling constructions, periodic boundary conditions and the counting of nodal domains. In particular, a new proof of Courant's nodal domain theorem is given, with an explicit formula for the nodal deficiency.

math.AP↗
arXiv2015

The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials

We study the Schrödinger operator $L=-Δ+V$ on a star-shaped domain $Ω$ in $\mathbb{R}^d$ with Lipschitz boundary $\partialΩ$. The operator is equipped with quite general Dirichlet- or Robin-type boundary conditions induced by operators between $H^{1/2}(\partialΩ)$ and $H^{-1/2}(\partialΩ)$, and the potential takes values in the set of symmetric $N\times N$ matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm-Lagrangian-Grassmannian of the subspace of $H^{1/2}(\partialΩ)\times H^{-1/2}(\partialΩ)$ corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of $L$ (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative squares of a bilinear form related to the boundary operator.

math.SP↗
arXiv2015

A Morse index theorem for elliptic operators on bounded domains

Given a selfadjoint, elliptic operator $L$, one would like to know how the spectrum changes as the spatial domain $Ω\subset \mathbb{R}^d$ is deformed. For a family of domains $\{Ω_t\}_{t\in[a,b]}$ we prove that the Morse index of $L$ on $Ω_a$ differs from the Morse index of $L$ on $Ω_b$ by the Maslov index of a path of Lagrangian subspaces on the boundary of $Ω$. This is particularly useful when $Ω_a$ is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the "original" problem (on $Ω_b$) and the "simplified" problem (on $Ω_a$). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.

math.AP↗
arXiv2014

The $L^2$ essential spectrum of the 2D Euler operator

Even in two dimensions, the spectrum of the linearized Euler operator is notoriously hard to compute. In this paper we give a new geometric calculation of the essential spectrum for 2D flows. This generalizes existing results---which are only available when the flow has arbitrarily long periodic orbits---and clarifies the role of individual streamlines in generating essential spectra.

math.AP↗
arXiv2014

Blowup solutions of Jang's equation near a spacetime singularity

We study Jang's equation on a one-parameter family of asymptotically flat, spherically symmetric Cauchy hypersurfaces in the maximally extended Schwarzschild spacetime. The hypersurfaces contain apparent horizons and are parametrized by their proximity to the singularity at $r = 0$. We show that on those hypersurfaces sufficiently close to the singularity, every radial solution to Jang's equation blows up. The proof depends only on the geometry in an arbitrarily small neighborhood of the singularity, suggesting that Jang's equation is in fact detecting the singularity. We comment on possible applications to the weak cosmic censorship conjecture.

gr-qc↗
arXiv2014

Estimation of CO$_2$ flux from targeted satellite observations: a Bayesian approach

We consider the estimation of carbon dioxide flux at the ocean-atmosphere interface, given weighted averages of the mixing ratio in a vertical atmospheric column. In particular we examine the dependence of the posterior covariance on the weighting function used in taking observations, motivated by the fact that this function is instrument-dependent, hence one needs the ability to compare different weights. The estimation problem is considered using a variational data assimilation method, which is shown to admit an equivalent infinite-dimensional Bayesian formulation. The main tool in our investigation is an explicit formula for the posterior covariance in terms of the prior covariance and observation operator. Using this formula, we compare weighting functions concentrated near the surface of the earth with those concentrated near the top of the atmosphere, in terms of the resulting covariance operators. We also consider the problem of observational targeting, and ask if it is possible to reduce the covariance in a prescribed direction through an appropriate choice of weighting function. We find that this is not the case---there exist directions in which one can never gain information, regardless of the choice of weight.

math.ST↗
arXiv2013

(Non)uniqueness of critical points in variational data assimilation

In this paper we apply the 4D-Var data assimilation scheme to the initialization problem for a family of quasilinear evolution equations. The resulting variational problem is non-convex, so it need not have a unique minimizer. We comment on the implications of non-uniqueness in numerical applications, then prove uniqueness results in the following situations: 1) the observational times are all sufficiently small; 2) the prior covariance is sufficiently small. We also give an example of a data set where the cost functional has a critical point of arbitrarily large Morse index.

math.OC↗
arXiv2012

Large-time uniqueness in a data assimilation problem for Burgers' equation

There is currently a great deal of interest in the 4D-Var data assimilation scheme, in which one uses observational data to find the optimal initial condition for a differential equation by minimizing a cost function over the set of all possible initial states. For nonlinear models this cost function can be nonconvex, and so the uniqueness of minimizers is not guaranteed. In this paper we apply 4D-Var to Burgers' equation and prove that, once a sufficient amount of data has been collected, there can be at most one physically reasonable minimizer to the variational problem.

math.OC↗