Higher Order Elliptic Operators Research Articles

Operator estimates for homogenization of higher-order elliptic operators with periodic coefficients

In L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) , a strongly elliptic differential operator A ε {\mathcal {A}}_\varepsilon of order 2 p 2p is studied. Its coefficients are periodic and depend on x / ε \mathbf {x}/\varepsilon . The resolvent ( A ε + I ) − 1 ({\mathcal {A}}_\varepsilon +I)^{-1} is approximated in the operator norm on L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) : ( A ε + I ) − 1 = ( A 0 + I ) − 1 + ∑ j = 1 2 p − 1 ε j K j , ε + O ( ε 2 p ) . \begin{equation*} ({\mathcal {A}}_\varepsilon +I)^{-1} = ({\mathcal {A}}^0+I)^{-1} + \sum _{j=1}^{2p-1} \varepsilon ^{j} {\mathcal {K}}_{j,\varepsilon } + O(\varepsilon ^{2p}). \end{equation*} Here A 0 {\mathcal {A}}^0 is an effective operator with constant coefficients, and K j , ε {\mathcal {K}}_{j,\varepsilon } , j = 1 , … , 2 p − 1 j=1,\dots ,2p-1 , are appropriate correctors.

Finsler-Rellich inequalities involving the distance to the boundary

We study Rellich inequalities associated to higher-order elliptic operators in the Euclidean space. The inequalities are expressed in terms of an associated Finsler metric. In the case of a half-space we obtain the sharp constant while for a general convex domain we obtain estimates that are better than those obtained by comparison with the polyharmonic operator.

Discrete SQG models with two boundaries and baroclinic instability of jet flows

In this paper, new vertically discrete versions of the surface quasigeostrophic (SQG) model with two boundaries are formulated. For any number of partition levels, the equations of the discrete model are written in the form of conservation laws for two Lagrangian invariants, which have the meaning of buoyancy distributions at the horizontal boundaries of the fluid layer. The values of the invariants are expressed in terms of the values of the stream function at two internal levels and contain higher order elliptic operators. The use of discrete models greatly simplifies the solution of problems of the linear theory of hydrodynamic stability and provides high accuracy even with a small number of vertical discrete levels. Using the two-level version of the SQG model, which is similar to the classical two–layer Phillips model, we investigated the linear stability of jet flows induced by piecewise constant boundary distributions of buoyancy. For these flows, analytical expressions for the growth rate of perturbations have been obtained and it is shown that the most unstable perturbation has a wavelength of the order of the Rossby baroclinic radius of deformation. Flows with vertical shear induced by smooth and slowly varying boundary buoyancy distributions are also considered. The instability of these flows is found to be absolute, that is, independent of the velocity profile horizontal structure.

Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients

Estimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L of higher order 2mge 4 have been shown so far only when the principal part of L is the polyharmonic operator (-Delta )^m. In the present note, it is shown that such kind of result still holds when the Laplacian is replaced by any second order uniformly elliptic operator in divergence form with smooth variable coefficients. For general higher order elliptic operators, whose principal part cannot be written as a power of second order operators, it was recently proved that such kind of result becomes false in general.

Higher-Order Operators on Networks: Hyperbolic and Parabolic Theory

We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators. We observe that they extend to the higher-order case and discuss well-posedness and conservation of energy of beam equations, along with regularizing properties of polyharmonic heat kernels. A noteworthy finding is the discovery of a new class of well-posed evolution equations with Wentzell-type boundary conditions.

Weak type (p,p) bounds for Schrödinger groups via generalized Gaussian estimates

Let L be a non-negative self-adjoint operator acting on L2(X), where X is a space of homogeneous type with a dimension n. Suppose that the heat operator e−tL satisfies the generalized Gaussian (p0,p0′)-estimates of order m for some 1≤p0<2. It is known that the operator (I+L)−seitL is bounded on Lp(X) for s≥n|1/2−1/p| and p∈(p0,p0′) (see for example, [5,7,9,10,13,26]). In this paper we study the endpoint case p=p0 and show that for s0=n|12−1p0|, the operator (I+L)−s0eitL is of weak type (p0,p0), that is, there is a constant C>0, independent of t and f so thatμ({x:|(I+L)−s0eitLf(x)|>α})≤C(1+|t|)n(1−p02)(‖f‖p0α)p0,t∈R for α>0 when μ(X)=∞, and α>(‖f‖p0/μ(X))p0 when μ(X)<∞. Our results can be applied to Schrödinger operators with rough potentials and higher order elliptic operators with bounded measurable coefficients although in general, their semigroups fail to satisfy Gaussian upper bounds.

Form-perturbation theory for higher-order elliptic operators and systems by singular potentials.

We give a form-perturbation theory by singular potentials for scalar elliptic operators on [Formula: see text] of order 2m with Hölder continuous coefficients. The form-bounds are obtained from an L1 functional analytic approach which takes advantage of both the existence of m-gaussian kernel estimates and the holomorphy of the semigroup in [Formula: see text] We also explore the (local) Kato class potentials in terms of (local) weak compactness properties. Finally, we extend the results to elliptic systems and singular matrix potentials. This article is part of the theme issue 'Semigroup applications everywhere'.

Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators

We study fundamental solutions of elliptic operators of order 2mge 4 with constant coefficients in large dimensions nge 2m, where their singularities become unbounded. For compositions of second order operators these can be chosen as convolution products of positive singular functions, which are positive themselves. As soon as nge 3, the polyharmonic operator (-Delta )^m may no longer serve as a prototype for the general elliptic operator. It is known from examples of Maz’ya and Nazarov (Math. Notes 39:14–16, 1986; Transl. of Mat. Zametki 39, 24–28, 1986) and Davies (J Differ Equ 135:83–102, 1997) that in dimensions nge 2m+3 fundamental solutions of specific operators of order 2mge 4 may change sign near their singularities: there are “positive” as well as “negative” directions along which the fundamental solution tends to +infty and -infty respectively, when approaching its pole. In order to understand this phenomenon systematically we first show that existence of a “positive” direction directly follows from the ellipticity of the operator. We establish an inductive argument by space dimension which shows that sign change in some dimension implies sign change in any larger dimension for suitably constructed operators. Moreover, we deduce for n=2m, n=2m+2 and for all odd dimensions an explicit closed expression for the fundamental solution in terms of its symbol. From such formulae it becomes clear that the sign of the fundamental solution for such operators depends on the dimension. Indeed, we show that we have even sign change for a suitable operator of order 2m in dimension n=2m+2. On the other hand we show that in the dimensions n=2m and n=2m+1 the fundamental solution of any such elliptic operator is always positive around its singularity.

Decay estimates for higher-order elliptic operators

This paper is mainly devoted to the study of time decay estimates of the higher-order Schrödinger-type operator H = ( − Δ ) m + V ( x ) H=(-\Delta )^{m}+V(x) in R n \mathbf {R}^{n} for n &gt; 2 m n&gt;2m and m ∈ N m\in \mathbf {N} . For certain decay potentials V ( x ) V(x) , we first derive the asymptotic expansions of resolvent R V ( z ) R_{V}(z) near zero threshold with the presence of zero resonance or zero eigenvalue, as well as identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen-type estimates and local decay estimates for higher-order Schrödinger propagator e − i t H e^{-itH} in the presence of zero resonance or zero eigenvalue. As a consequence, the endpoint Strichartz estimate and L p L^{p} -decay estimates can also be obtained. Finally, by a virial argument, a criterion on the absence of positive embedding eigenvalues is given for ( − Δ ) m + V ( x ) (-\Delta )^{m}+V(x) with a repulsive potential.

Uniform boundary estimates in homogenization of higher-order elliptic systems

This paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators $$\mathcal {L}_\varepsilon $$ , with rapidly oscillating periodic coefficients. We derive uniform boundary $$C^{m-1,\lambda } (0\!<\!\lambda \!<\!1)$$ and $$ W^{m,p}$$ estimates in $$C^1$$ domains, as well as uniform boundary $$C^{m-1,1}$$ estimate in $$C^{1,\theta } (0\!<\!\theta \!<\!1)$$ domains without the symmetry assumption on the operator. The proof, motivated by the works “Armstrong and Smart in Ann Sci Ec Norm Super (4) 49(2):423–481 (2016) and Shen in Anal PDE 8(7):1565–1601 (2015),” is based on a suboptimal convergence rate in $$H^{m-1}(\Omega )$$ . Compared to “Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036 (2012) and Shen (2015),” the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of $$u_0$$ (the solution to the homogenized problem), and thus might be of some independent interests even for second-order elliptic systems.

Sparse operator compression of higher-order elliptic operators with rough coefficients

We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy minimizing functions on local patches. On a regular mesh with mesh size h, the localized basis functions have supports of diameter O(hlog (1/h)) and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter O(hlog (1/h)), our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate O(h^k) for a (2k)th-order elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Matérn covariance functions can be approximated by a rank-n operator with a localized basis and with the optimal accuracy.

Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

Let L = L 0 + V be a Schrodinger type operator, where L 0 is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L q boundedness of Riesz transform ∇ m L −1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e −t L . Furthermore, the authors impose extra regularity assumptions on V to obtain the L q boundedness of Riesz transform ∇ m L −1/2 for some q > 2. In particular, these results are applied to the more interesting Schrodinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients.

Local Hardy spaces associated with inhomogeneous higher order elliptic operators

Let [Formula: see text] be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all [Formula: see text] and [Formula: see text] satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces [Formula: see text] associated with [Formula: see text], which coincide with Goldberg’s local Hardy spaces [Formula: see text] for all [Formula: see text] when [Formula: see text] (the Laplace operator). The authors also establish a real-variable theory of [Formula: see text], which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when [Formula: see text] (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that [Formula: see text] coincides with the Hardy space [Formula: see text] associated with the operator [Formula: see text] for all [Formula: see text], where [Formula: see text] is some positive constant depending on the ellipticity and the off-diagonal estimates of [Formula: see text]. As an application, the authors establish some mapping properties for the local Riesz transforms [Formula: see text] on [Formula: see text], where [Formula: see text] and [Formula: see text].

Operator estimates in homogenization theory

This paper gives a systematic treatment of two methods for obtaining operator estimates: the shift method and the spectral method. Though substantially different in mathematical technique and physical motivation, these methods produce basically the same results. Besides the classical formulation of the homogenization problem, other formulations of the problem are also considered: homogenization in perforated domains, the case of an unbounded diffusion matrix, non-self- adjoint evolution equations, and higher-order elliptic operators. Bibliography: 62 titles.

Estimates in homogenization of higher-order elliptic operators

A divergent-type elliptic operator of arbitrary even order 2m is studied. Coefficients of the operator are -periodic, is a small parameter. The resolvent equation is solvable in the Sobolev space of order m for any , provided the parameter is sufficiently large, , where the bound depends only on constants from ellipticity condition. The limit equation is of the same type but with constant coefficients, that is, . The limit operator can be considered here, for instance, in the sense of G-convergence. We prove that the resolvent approximates in operator -norm with the estimate , as . We find also the approximation of the resolvent in operator -norm. This is the sum , where is a correcting operator whose structure is given. We prove the estimate , as .

Неравенство Гординга для эллиптических операторов высшего порядка с нестепенным вырождением и его приложения

Неравенство Гординга для эллиптических операторов высшего порядка с нестепенным вырождением и его приложения

Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators

Abstract Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and ( p - ⁢ ( L ) , p + ⁢ ( L ) ) ${(p_{-}(L),p_{+}(L))}$ be the maximal interval of exponents q ∈ [ 1 , ∞ ] ${q\in[1,\infty]}$ such that the semigroup { e - t ⁢ L } t &gt; 0 ${\{e^{-tL}\}_{t&gt;0}}$ is bounded on L q ⁢ ( ℝ n ) ${L^{q}(\mathbb{R}^{n})}$ . In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces H L p ⁢ ( ℝ n ) ${H_{L}^{p}(\mathbb{R}^{n})}$ for all p ∈ ( 0 , p + ⁢ ( L ) ) ${p\in(0,p_{+}(L))}$ , which when p = 1 ${p=1}$ , answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize H L p ⁢ ( ℝ n ) ${H_{L}^{p}(\mathbb{R}^{n})}$ via various versions of square functions and Lusin-area functions associated to the operator L.

Spectral stability results for higher-order operators under perturbations of the domain

We analyze the spectral behavior of higher-order elliptic operators when the domain is perturbed. We provide general spectral stability results for Dirichlet and Neumann boundary conditions. Moreover, we study the bi-harmonic operator with the so-called intermediate boundary conditions. We give special attention to this last case and analyze its behavior when the boundary of the domain has some oscillatory behavior. We will show that there is a critical oscillatory behavior and that the limit problem depends on whether we are above, below or just sitting on this critical value.

A free boundary problem for higher order elliptic operators

Let be a domain in and suppose, locally, there exists a function which, in the sense of distributions, solves the problem where is the unit ball in and is a fourth order elliptic operator, i.e. for With no sign assumption on and by using blow-up techniques we will prove that is in

Gårding’s inequality for higher order elliptic operators with nonpower degeneration

Gårding’s inequality for higher order elliptic operators with nonpower degeneration