Elliptic Partial Differential Operators Research Articles

Spectral asymptotics of elliptic operators on manifolds

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator [Formula: see text] directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function [Formula: see text] and the heat trace [Formula: see text]. The kernel [Formula: see text] of the heat semigroup [Formula: see text], called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as [Formula: see text], that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as [Formula: see text], that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

Analyticity of parametric elliptic eigenvalue problems and applications to quasi-Monte Carlo methods

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion a 0 ( x ) + ∑ j ∈ N y j a j ( x ) , where elements of the parameter vector y = ( y j ) j ∈ N ∈ U ∞ are independent and identically uniformly distributed on U := [ − 1 2 , 1 2 ] . Under the assumption ‖ ∑ j ∈ N ρ j | a j | ‖ L ∞ ( D ) < ∞ with some positive sequence ( ρ j ) j ∈ N ∈ ℓ p ( N ) for p ∈ ( 0 , 1 ] we show that for any y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues ( λ j ( y ) ) j ∈ N which can be ordered non-decreasingly. Moreover, the spectral gap λ 2 ( y ) − λ 1 ( y ) is uniformly positive in U ∞ . From this, we prove the holomorphic extension property of λ 1 ( y ) to a complex domain in C ∞ and estimate partial derivatives of λ 1 ( y ) with respect to the parameter y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of λ 1 ( y ) .

Comparison principles for second-order elliptic/parabolic equations with discontinuities in the gradient compatible with Finsler norms

This paper is about elliptic and parabolic partial differential operators with discontinuities in the gradient which are compatible with a Finsler norm in a sense to be made precise. Examples of this type of problems arise in a number of contexts, most notably the recent work of Chatterjee and the second author [7] on scaling limits of discrete surface growth models as well as L∞−variational problems. Building on the approach of Ishii [16], new comparison results are proven within a unified framework that includes a number of previous results as special cases.

Super-localization of elliptic multiscale problems

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a d d -dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter H H . This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximation space. This paper presents a novel localization technique that leads to a super-exponential decay of its basis relative to H H . This suggests that basis functions with supports of width O ( H | log ⁡ H | ( d − 1 ) / d ) \mathcal O(H|\log H|^{(d-1)/d}) are sufficient to preserve the optimal algebraic rates of convergence in H H without pre-asymptotic effects. A sequence of numerical experiments illustrates the significance of the new localization technique when compared to the so far best localization to supports of width O ( H | log ⁡ H | ) \mathcal O(H|\log H|) .

A Liouville comparison principle for semilinear elliptic second-order inequalities

We consider semilinear elliptic second-order partial differential inequalities of the form (*) in the whole space , where , q>0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are measurable and locally bounded such that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of capacities associated with the operator L for solutions of (*) which are measurable and belong locally in to a Sobolev-type function space also associated with the operator L.

Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space

We consider a locally uniformly strictly elliptic second order partial differential operator in Rd, d≥2, with low regularity assumptions on its coefficients, as well as an associated Hunt process and semigroup. The Hunt process is known to solve a corresponding stochastic differential equation that is pathwise unique. In this situation, we study the relation of invariance, infinitesimal invariance, recurrence, transience, conservativeness and Lr-uniqueness, and present sufficient conditions for non-existence of finite infinitesimally invariant measures as well as finite invariant measures. Our main result is that recurrence implies uniqueness of infinitesimally invariant measures, as well as existence and uniqueness of invariant measures, both in subclasses of locally finite measures. We can hence make in particular use of various explicit analytic criteria for recurrence that have been previously developed in the context of (generalized) Dirichlet forms and present diverse examples and counterexamples for uniqueness of infinitesimally invariant, as well as invariant measures and an example where L1-uniqueness fails for one infinitesimally invariant measure but holds for another and pathwise uniqueness holds. Furthermore, we illustrate how our results can be applied to related work and vice versa.

Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps

Let $\\Omega \\subset \\mathbb{R}^d$ be a bounded open set with Lipschitz boundary $\\Gamma$. It will be shown that the Jordan chains of m-sectorial second-order elliptic partial differential operators with measurable coefficients and (local or non-local) Robin boundary conditions in $L_2(\\Omega)$ can be characterized with the help of Jordan chains of the Dirichlet-to-Neumann map and the boundary operator from $H^{1/2}(\\Gamma)$ into $H^{-1/2}(\\Gamma)$. This result extends the Birman–Schwinger principle in the framework of elliptic operators for the characterization of eigenvalues, eigenfunctions and geometric eigenspaces to the complete set of all generalized eigenfunctions and algebraic eigenspaces.

Gaussian fields, equilibrium potentials and multiplicative chaos for Dirichlet forms

For a Dirichlet form $(\mathcal {E},\mathcal {F})$ on L2(E;m), let $\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}$ be the Gaussian field indexed by the extended Dirichlet space $\mathcal {F}_{e}$. We first solve the equilibrium problem for a regular recurrent Dirichlet form $\mathcal {E}$ of finding for a closed set B a probability measure μB concentrated on B whose recurrent potential $R\mu ^{B}\in \mathcal {F}_{e}$ is constant q.e. on B (called a Robin constant). We next assume that E is the complex plane $\mathbb {C}$ and $\mathcal {E}$ is a regular recurrent strongly local Dirichlet form. For the closed disk $\bar B(\textbf {x},r)=\{\textbf {z}\in \mathbb {C}:|\textbf {z}-\textbf {x}|\le r\}$, let μx, r and f(x, r) be its equilibrium measure and Robin constant. Denote the Gaussian random variable $X_{R\mu ^{\textbf {x}.r}}\in \mathbb {G}(\mathcal {E})$ by Yx, r and let, for a given constant γ > 0, $\mu _{r}(A,\omega )={\int \limits }_{A} \exp (\gamma Y^{\textbf {x},r}-(1/2)\gamma ^{2} f(\textbf {x},r))d\textbf {x}.$ Under a certain condition on the growth rate of f(x, r), we prove the convergence in probability of μr(A, ω) to a random measure $\overline {\mu }(A,\omega )$ as r↓ 0. The possible range of γ to admit a non-trivial limit will then be examined in the cases that $(\mathcal {E}.\mathcal {F})$ equals $(\frac 12{\textbf {D}}_{\mathbb {C}},H^{1}(\mathbb {C}))$ and $(\textbf {a},H^{1}(\mathbb {C}))$, where a corresponds to the uniformly elliptic partial differential operator of divergence form.

Existence and regularity of infinitesimally invariant measures, transition functions and time-homogeneous Itô-SDEs

We show existence of an infinitesimally invariant measure $m$ for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some local integrability order. Subsequently, we derive regularity properties of the corresponding semigroup which is defined in $L^s(\mathbb{R}^d,m)$, $s\in [1,\infty]$, including the classical strong Feller property and classical irreducibility. This leads to a transition function of a Hunt process that is explicitly identified as a solution to an SDE. Further properties of this Hunt process, like non-explosion, moment inequalities, recurrence and transience, as well as ergodicity, including invariance and uniqueness of $m$, and uniqueness in law, can then be studied using the derived analytical tools and tools from generalized Dirichlet form theory.

Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations

We develop strong and weak maximum principles for boundary-degenerate elliptic and parabolic linear second-order partial differential operators, A u := − t r ( a D 2 u ) − ⟨ b , D u ⟩ + c u Au := -\mathrm {tr}(aD^2u)-\langle b, Du\rangle + cu , with partial Dirichlet boundary conditions. The coefficient, a ( x ) a(x) , is assumed to vanish along a nonempty open subset, ∂ 0 O \partial _0\mathscr {O} , called the degenerate boundary portion, of the boundary, ∂ O \partial \mathscr {O} , of the domain O ⊂ R d \mathscr {O}\subset \mathbb {R}^d , while a ( x ) a(x) is nonzero at any point of the nondegenerate boundary portion, ∂ 1 O := ∂ O ∖ ∂ 0 O ¯ \partial _1\mathscr {O} := \partial \mathscr {O}\setminus \overline {\partial _0\mathscr {O}} . If an A A -subharmonic function, u u in C 2 ( O ) C^2(\mathscr {O}) or W l o c 2 , d ( O ) W^{2,d}_{\mathrm {loc}}(\mathscr {O}) , is C 1 C^1 up to ∂ 0 O \partial _0\mathscr {O} and has a strict local maximum at a point in ∂ 0 O \partial _0\mathscr {O} , we show that u u can be perturbed, by the addition of a suitable function w ∈ C 2 ( O ) ∩ C 1 ( R d ) w\in C^2(\mathscr {O})\cap C^1(\mathbb {R}^d) , to a strictly A A -subharmonic function v = u + w v=u+w having a local maximum in the interior of O \mathscr {O} . Consequently, we obtain strong and weak maximum principles for A A -subharmonic functions in C 2 ( O ) C^2(\mathscr {O}) and W l o c 2 , d ( O ) W^{2,d}_{\mathrm {loc}}(\mathscr {O}) which are C 1 C^1 up to ∂ 0 O \partial _0\mathscr {O} . Points in ∂ 0 O \partial _0\mathscr {O} play the same role as those in the interior of the domain, O \mathscr {O} , and only the nondegenerate boundary portion, ∂ 1 O \partial _1\mathscr {O} , is required for boundary comparisons. Moreover, we obtain a comparison principle for a solution and supersolution in W l o c 2 , d ( O ) W^{2,d}_{\mathrm {loc}}(\mathscr {O}) to a unilateral obstacle problem defined by A A , again where only the nondegenerate boundary portion, ∂ 1 O \partial _1\mathscr {O} , is required for boundary comparisons. Our results extend those of Daskalopoulos and Hamilton, Epstein and Mazzeo, and Feehan, where t r ( a D 2 u ) \mathrm {tr}(aD^2u) is in addition assumed to be continuous up to and vanish along ∂ 0 O \partial _0\mathscr {O} in order to yield comparable maximum principles for A A -subharmonic functions in C 2 ( O ) C^2(\mathscr {O}) , while the results developed here for A A -subharmonic functions in W l o c 2 , d ( O ) W^{2,d}_{\mathrm {loc}}(\mathscr {O}) are entirely new. Finally, we obtain analogues of all the preceding results for parabolic linear second-order partial differential operators, L u := − u t − t r ( a D 2 u ) − ⟨ b , D u ⟩ + c u Lu := -u_t - \mathrm {tr}(aD^2u)-\langle b, Du\rangle + cu .

Bogolyubov invariant via relative spectral invariants on manifolds

We introduce and study a new spectral invariant of two elliptic partial differential operators of Laplace and Dirac types on compact smooth manifolds without boundary that depends on both the eigenvalues and the eigensections of these operators, which is equal in some approximation to the regularized number of created particles from the vacuum when the dynamical operator depends on time. We study the asymptotic expansion of this invariant for small adiabatic parameter and compute explicitly the first two coefficients of the asymptotic expansion.

The algebra of Wick polynomials of a scalar field on a Riemannian manifold

On a connected, oriented, smooth Riemannian manifold without boundary we consider a real scalar field whose dynamics is ruled by [Formula: see text], a second-order elliptic partial differential operator of Laplace type. Using the functional formalism and working within the framework of algebraic quantum field theory and of the principle of general local covariance, first we construct the algebra of locally covariant observables in terms of equivariant sections of a bundle of smooth, regular polynomial functionals over the affine space of the parametrices associated to [Formula: see text]. Subsequently, adapting to the case in hand a strategy first introduced by Hollands and Wald in a Lorentzian setting, we prove the existence of Wick powers of the underlying field, extending the procedure to smooth, local and polynomial functionals and discussing in the process the regularization ambiguities of such procedure. Subsequently we endow the space of Wick powers with an algebra structure, dubbed E-product, which plays in a Riemannian setting the same role of the time-ordered product for field theories on globally hyperbolic spacetimes. In particular, we prove the existence of the E-product and we discuss both its properties and the renormalization ambiguities in the underlying procedure. As the last step, we extend the whole analysis to observables admitting derivatives of the field configurations and we discuss the quantum Møller operator which is used to investigate interacting models at a perturbative level.

Relative spectral invariants of elliptic operators on manifolds

We introduce and study new relative spectral invariants of two elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depend on both the eigenvalues and the eigensections of these operators and contain much more information about geometry. We prove the existence of the homogeneous short time asymptotics of the new invariants with the coefficients of the asymptotic expansion being integrals of some invariants that depend on the symbols of both operators. The first two coefficients of the asymptotic expansion are computed explicitly.

Convergence Rates for Penalized Least Squares Estimators in PDE Constrained Regression Problems

We consider PDE constrained nonparametric regression problems in which the parameter $f$ is the unknown coefficient function of a second order elliptic partial differential operator $L_f$, and the ...

Sampling and equidistribution theorems for elliptic second order operators, lifting of eigenvalues, and applications

We consider elliptic second order partial differential operators with Lipschitz continuous leading order coefficients on finite cubes and the whole Euclidean space. We prove quantitative sampling and equidistribution theorems for eigenfunctions. The estimates are scale-free, in the sense that for a sequence of growing cubes we obtain uniform estimates. These results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum, and an uncertainty relation (aka spectral inequality) for short energy interval spectral projectors. Several applications including random operators are discussed. In the proof we have to overcome several challenges posed by the variable coefficients of the leading term.

First-Kind Boundary Integral Equations for the Hodge--Helmholtz Operator

We adapt the variational approach to the analysis of first-kind boundary integral equations associated with strongly elliptic partial differential operators from [M. Costabel, SIAM J. Math. Anal., ...

On the summability of the spectral expansions associated with the elliptic differential operators

In this paper we study the summability problems of the spectral expansions associated with the elliptic partial differential operators in the spaces of distributions. In particular, the problems studied for the Schrodinger operator with the singular potential. For this operator theorems on the summability of the Fourier expansions in the generalised Holder spaces and in the Sobolev spaces of the mixed norm proved. The conditions obtained are accurate in the classes of considered distributions.

Flux reconstructions in the Lehmann–Goerisch method for lower bounds on eigenvalues

The standard application of the Lehmann–Goerisch method for lower bounds on eigenvalues of symmetric elliptic second-order partial differential operators relies on determination of fluxes σ̃i that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes σ̃i are usually computed by solving a global saddle point problem with mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes σ̃i of the same quality. The simplified problem is smaller, it is positive definite, and any H(div,Ω) conforming finite elements, such as Raviart–Thomas elements, can be used for its solution. In addition, these global problems can be split into a number of independent local problems on patches, which allows for trivial parallelization. The computational performance of these approaches is illustrated by numerical examples for Laplace and Steklov type eigenvalue problems. These examples also show that local flux reconstructions enable computation of lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions.

Asymptotic analysis of a kernel estimator for parabolic stochastic partial differential equations driven by fractional noises

We study a strongly elliptic partial differential operator with time-varying coeffcient in a parabolic diagonalizable stochastic equation driven by fractional noises. Based on the existence and uniqueness of the solution, we then obtain a kernel estimator of time-varying coeffcient and the convergence rates. An example is given to illustrate the theorem.

Analytic dependence of a periodic analog of a fundamental solution upon the periodicity parameters

We prove an analyticity result in Sobolev–Bessel potential spaces for the periodic analog of the fundamental solution of a general elliptic partial differential operator upon the parameters which determine the periodicity cell. Then we show concrete applications to the Helmholtz and the Laplace operators. In particular, we show that the periodic analogs of the fundamental solution of the Helmholtz and of the Laplace operator are jointly analytic in the spatial variable and in the parameters which determine the size of the periodicity cell. The analysis of the present paper is motivated by the application of the potential theoretic method to periodic anisotropic boundary value problems in which the “degree of anisotropy” is a parameter of the problem.