Order Elliptic Operators Research Articles

Existence and Stability of Solutions of General Semilinear Elliptic Equations with Measure Data

Abstract We study existence and stability for solutions of −Lu + g(x, u) = ω where L is a second order elliptic operator, g a Caratheodory function and ω a measure in Ω. We present a unified theory of the Dirichlet problem and the Poisson equation. We prove the stability of the problem with respect to weak convergence of the data.

Gaussian upper bounds for heat kernels of second order complex elliptic operators with unbounded diffusion coefficients on arbitrary domains

In this paper we obtain Gaussian upper bounds for the integral kernel of the semigroup associated with second order elliptic differential operators with complex unbounded measurable coefficients defined in a domain Ω of ℝ N and subject to various boundary conditions. In contrast to the previous literature the diffusions coefficients are not required to be bounded or regular. A new approach based on Davies-Gaffney estimates is used. It is applied to a number of examples, including degenerate elliptic operators arising in Financial Mathematics and generalized Ornstein-Uhlenbeck operators with potentials.

A generalization of Douglis–Nirenberg elliptic operators

It is well known that the composition of two elliptic differential operators of orders ω and ω′ is again an elliptic operator of order ω+ω′. This fact is not true for the composition of two multi–order systems elliptic in the sense of Douglis and Nirenberg. We consider a new more general class of multi–order pseudodifferential operators. This class of multi–order essentially elliptic operators is closed with respect to the composition, coordinate and basis transformation. Moreover, every multi–order essentially elliptic operator has a parametrix belonging to the class and therefore is a Fredholm operator. Bibliography: 5 titles.

Elliptic Variational Inequalities with Discontinuous Multi-Valued Lower Order Terms

AbstractWe consider multi-valued elliptic variational inequalities for operators of the formu ↦ Au + ∂2ψ(u, u),where A is a second order elliptic operator of Leray-Lions type, and u ↦ ∂2ψ(u, u) is a multi-valued lower order term that may neither be lower nor upper semicontinuous, see e.g., Figure 2. More precisely, the lower order term is generated by some function (r, s) → ψ(r, s) with ψ : ℝ × ℝ → ℝ that is locally Lipschitz with respect to its second variable s for each r ∈ ℝ, and ∂2ψ(r, s) denotes Clarke’s generalized gradient of s ↦ ψ(r, s). The novelty of this paper is that the multifunction r ↦ ∂2ψ(r, s) may discontinuously depend on r in a certain specified way, which gives rise to the new class of discontinuous multi-valued lower order terms s ↦ ∂2ψ(s, s). Though rich in structure, the characteristic features of this new class of multi-valued lower order terms can easily be described and verified. Our main goal is to provide an analytical framework and to prove existence and comparison results for this new class of multi-valued variational inequalities that includes the theory of variational-hemivariational inequalities as special case.

Bounds on the non-real spectrum of differential operators with indefinite weights

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and $$\infty $$ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm–Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.

An $H_\mathsf{div}$-Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems

This work considers the Cauchy problem for a second order elliptic operator in a bounded domain. A new quasi-reversibility approach is introduced for approximating the solution of the ill-posed Cauchy problem in a regularized manner. The method is based on a well-posed mixed variational problem on $H^1\times H_\mathsf{div}$ with the corresponding solution pair converging monotonically to the solution of the Cauchy problem and the associated flux, if they exist. It is demonstrated that the regularized problem can be discretized using Lagrange and Raviart--Thomas finite elements. The functionality of the resulting numerical algorithm is tested via three-dimensional numerical experiments based on simulated data. Both the Cauchy problem and a related inverse obstacle problem for the Laplacian are considered.

Eigenvalue multiplicities for second order elliptic operators on networks

Eigenvalue multiplicities for second order elliptic operators on networks

Partial Cauchy Data for General Second Order Elliptic Operators in Two Dimensions

We consider the inverse problem of determining the coefficients of a general second order elliptic operator in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. We show that one can determine the coefficients of the operator up to natural limitations such as conformal invariance, gauge transformations and diffeomorphism invariance. We use the main result to prove that the curl of the magnetic field and the electric potential are uniquely determined by measuring partial Cauchy data associated to the magnetic Schrödinger equation on an arbitrary open subset of the boundary. We also show that any two of the three coefficients of a second order elliptic operator whose principal part is the Laplacian, are uniquely determined by their partial Cauchy data.

Efficient Rearrangement Algorithms for Shape Optimization on Elliptic Eigenvalue Problems

In this paper, several efficient rearrangement algorithms are proposed to find the optimal shape and topology for elliptic eigenvalue problems with inhomogeneous structures. The goal is to solve minimization and maximization of the k-th eigenvalue and maximization of spectrum ratios of the second order elliptic differential operator. Physically, these problems are motivated by the frequency control based on density distribution of vibrating membranes. The methods proposed are based on Rayleigh quotient formulation of eigenvalues and rearrangement algorithms which can handle topology changes automatically. Due to the efficient rearrangement strategy, the new proposed methods are more efficient than classical level set approaches based on shape and/or topological derivatives. Numerous numerical examples are provided to demonstrate the robustness and efficiency of new approach.

Characterizations of Hardy spaces associated to higher order elliptic operators

In this paper, the authors first show that the classical Hardy space H1(Rn) can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups e−tP and e−tP, respectively, where P is an elliptic operator with real constant coefficients of homogeneous order 2m (m⩾1). Moreover, the authors also prove that H1(Rn) can be characterized by the Riesz transforms ∇mP−1/2 if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space HL1(Rn) and give its characterizations by area integrals related to the semigroups e−tL and e−tL, respectively. Finally, authors give the (HL1,L1) boundedness of Riesz transforms, square functions and maximal functions associated with L.

Spectral stability estimates for the eigenfunctions of second order elliptic operators

AbstractStability of the eigenfunctions of nonnegative selfadjoint second‐order linear elliptic operators subject to homogeneous Dirichlet boundary data under domain perturbation is investigated. Let Ω, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega ^{\prime } \subset \mathbb {R}^n$\end{document} be bounded open sets. The main result gives estimates for the variation of the eigenfunctions under perturbations Ω′ of Ω such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega _{\varepsilon } = \lbrace x \in \Omega : {\rm dist}(x,\, \mathbb {R}^n \!\setminus \! \Omega ) > \varepsilon \rbrace \subset \Omega ^{\prime } \subset \overline{\Omega ^{\prime }} \subset \Omega$\end{document} in terms of powers of ε, where the parameter ε > 0 is sufficiently small. The estimates obtained here hold under some regularity assumptions on Ω, Ω′. They are obtained by using the notion of a gap between linear operators, which has been recently extended by the authors to differential operators defined on different open sets.

Optimal Hardy-type inequalities for elliptic operators

For a general second order elliptic operator P in a domain Ω, we construct a Hardy weight W in the punctured domain Ω⋆:=Ω∖{0} such that P−λW is subcritical in Ω⋆ for λ<1, null-critical in Ω⋆ for λ=1, and supercritical near infinity and near 0 for λ>1. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy weight is given by an explicit formula involving the Green function of P and its gradient.

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

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

On positive harmonic functions in cones and cylinders

We first consider a question raised by Alexander Eremenko and show that if \Omega is an arbitrary connected open cone in \mathbb{R}^d , then any two positive harmonic functions in \Omega that vanish on \partial \Omega must be proportional – an already known fact when \Omega has a Lipschitz basis or more generally a John basis. It is also shown however that when d \geq 4 , there can be more than one Martin point at infinity for the cone though non-tangential convergence to the canonical Martin point at infinity always holds. In contrast, when d \leq 3 , the Martin point at infinity is unique for every cone. These properties connected with the dimension are related to well-known results of M. Cranston and T. R. McConnell about the lifetime of conditioned Brownian motions in planar domains and also to subsequent results by R. Bañuelos and B. Davis. We also investigate the nature of the Martin points arising at infinity as well as the effects on the Martin boundary resulting from the existence of John cuts in the basis of the cone or from other regularity assumptions. The main results together with their proofs extend to cylinders \mathcal{C}_Y(\Sigma )= { \mathbb R} \times \Sigma (where \Sigma is a relatively compact region of a manifold M ), equipped with a suitable second order elliptic operator.

Sufficient Conditions of Optimality for Convex Differential Inclusions of Elliptic Type and Duality

This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic type. On the basis of conjugacy correspondence the dual problems are constructed. Using the new concepts of locally adjoint mappings in the form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality problems are formulated for convex problems and duality theorems are proved. The results obtained are generalized to the multidimensional case with a second order elliptic operator.

On the regularity of solutions of p-adic parabolic equations

In this article we study the following Cauchy problem: Open image in new window here F(D, α) is an elliptic operator of order d, g(x, t) is a locally constant function with respect to x and infinitely differentiable with respect to t, for t > 0, and φ is in the domain of F(D, α). We show that the above initial value problem has a unique solution u(x, t), which is infinitely differentiable with respect to t, and has Taibleson derivatives up to order γ ≤ αd − n.

On continuity properties of monotone operators beyond the natural domain of definition

We study boundary value problems associated to a nonlinear elliptic system of partial differential equations. The leading second order elliptic operator provides L2-coerciveness and has at most linear growth with respect to the gradient. Incorporating properties of Lipschitz approximations of Sobolev functions we are able to show that the problems in consideration are well-posed, in the sense of Hadamard, in W1,p for all \({p \in (p_0,2]}\) for certain \({p_0 \in [3/2,2)}\) . For simplicity we restrict ourselves to the Dirichlet problem.

Abstract elliptic and parabolic systems with applications to problems in cylindrical domains

We consider problems for quite general second-order abstract elliptic and corresponding parabolic equations on the interval $[0,1]$ and the rectangle $[0,T]\times [0,1]$, respectively. $R$-boundedness estimates of solutions of abstract boundary-value problems for elliptic equations with a parameter are established, in contrast to standard norm-bounded estimates. The results are applied to obtain $L^p$-maximal regularity for corresponding parabolic systems. In applications, the coefficient $A(x)$ of the solution $u$ can be $2m$-order elliptic operators with suitable boundary conditions, while the coefficient $B(x)$ of the first-order derivative of the solution $D_xu$ can be interpreted as an $m$-order differential operator. The corresponding applications to PDEs are presented.

On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ t μ = L * μ for bounded Borel measures on ℝ d × (0, T), where L is a second order elliptic operator, for example, $$ Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) $$ , and the equation is understood as the identity $$ \int \left( {{\partial_t}u + Lu} \right)d\mu = 0 $$ for all smooth functions u with compact support in ℝ d × (0, T). Our study are motivated by equations of such a type, namely, the Fokker–Planck–Kolmogorov equations for transition probabilities of diffusion processes. Solutions are considered in the class of probability measures and in the class of signed measures with integrable densities. We present some recent positive results, give counterexamples, and formulate open problems. Bibliography: 34 titles.

Spatial pseudoanalytic functions arising from the factorization of linear second order elliptic operators

Biquaternionic Vekua-type equations arising from the factorization of linear second order elliptic operators are studied. Some concepts from classical pseudoanalytic function theory are generalized onto the considered spatial case. The derivative and antiderivative of a spatial pseudoanalytic function are introduced and their applications to the second order elliptic equations are considered. Copyright © 2011 John Wiley & Sons, Ltd.