Journal of Operators

Journal of Operators / 2013 / Article

Review Article | Open Access

Volume 2013 |Article ID 562963 | https://doi.org/10.1155/2013/562963

Denise Huet, "A Survey of Some Topics Related to Differential Operators", Journal of Operators, vol. 2013, Article ID 562963, 17 pages, 2013. https://doi.org/10.1155/2013/562963

A Survey of Some Topics Related to Differential Operators

Academic Editor: Qingkai Kong
Received26 Mar 2013
Revised02 Oct 2013
Accepted04 Oct 2013
Published18 Dec 2013

Abstract

This paper is the result of investigations suggested by recent publications and completes the work of Huet, 2010. The topics, which are dealt with, concern some spaces of functions and properties of solutions of linear and nonlinear, stationary and evolution differential equations, namely, existence, spectral properties, resonances, singular perturbations, boundary layers, and inertial manifolds. They are presented in the alphabetical order. The aim of this document and of Huet, 2010, is to be a useful reference for (young) researchers in mathematics and applied sciences.

1. Introduction

The article is divided into several sections entitled: Birman-Schwinger operators; BMO spaces; Bounded variation (functions of); Discrete energy; Dissipative operators; Dynamical systems; Equal- area condition; Inertial manifolds; Mathieu-Hill type equations; Memory (equations with); Nodes, Nodal; Resonances. The development of each entry includes indications on history, definitions, an overview of main results, examples, and applications but is, of course, nonexhaustive. Complements will be found in the references. A prepublication of some entries is presented in Huet [1].

2. Birman-Schwinger Operator

Definition 1. Consider the Schrödinger operator acting on , , where is a real-valued continuous function defined on which is nonnegative and tends to zero, sufficiently fast, as   and   is a small negative coupling constant. The operator is self-adjoint and its spectrum is . The Birman-Schwinger operator associated with (1) is the operator where is the resolvent of in . For each ,   is self-adjoint and compact (cf. Arazy and Zelenko [2]).

Application. In [2], the authors consider the decomposition , where is a finite rank operator and an Hilbert-Schmidt operator whose norm is uniformly bounded with respect to for some . An asymptotic expansion of the bottom virtual eigenvalue of , as tends to zero, is deduced from this decomposition: if is odd, it is of power type, while, when is even, it involves the log function. Asymptotic estimates are obtained, as , for the nonbottom virtual eigenvalues of , , where if is odd and if is even. If is odd, is a meromorphic operator function, and the leading terms of the asymptotic estimates of are of power type. An algorithm, based on the Puiseux-Newton diagram (cf. Baumgärtel [3]), is proposed for an evaluation of the leading coefficients of these estimates. If d is even, two-sided estimates are obtained for eigenvalues with an exponential rate of decay; the rest of the eigenvalues have a power rate of decay. Estimates of Lieb-Thirring type are obtained for groups of eigenvalues which have the same rate of decay, when is odd or even.

3. BMO Spaces

3.1. BMO and Related Spaces

Definition 2. Let be the supremum over all cubes with edges parallel to the coordinate axes, the sidelength of , and the mean value of over . The John-Nirenberg’s space (cf. John and Nirenberg [4]) is the space of locally integrable complex-valued functions defined on , such that The space of functions of bounded mean oscillation, modulo constants, equipped with the above norm, is a Banach space.

Definition 3 (real Hardy space ). A function if and only if where , , each function is supported on a ball and has integral zero, and . Functions that satisfy the above properties are called 1-atom (cf. Stein, [5]). With the norm where the inf is taken on all decompositions of of the form (4), is a Banach space. In Fefferman and Stein [6, Theorem 2], it is proved that the dual of is . For definitions and properties of Hardy-spaces , see [6].
Related spaces are presented in [7].

3.2. BMO Nonlinearity (cf. Byun and Wang [8])

Let be a nonlinear, real valued, function on . For and , denotes the open ball of radius centered at . Set and define the function by

Definition 4 (-BMO condition). The function satisfies the -BMO condition if

Definition 5 (Reifenberg domain). A bounded open set in is -Reifenberg flat if, for every and every , there exists a coordinate system (which can depend on and ) so that is the origin in this coordinate system and that (cf. [9]).

Applications. (i) Let be a bounded, open subset of , , , is a vector field measurable in for almost every and continuous in for each . In [8], the authors consider the nonlinear boundary value problem The following conditions are imposed on : for all and almost every , for all and almost , and for some positive constants , ,  and  . Then it is proved that there exists such that, if a satisfies the -BMO condition and is -Reifenberg flat, the weak solution to (10) belongs to with the estimate where is independent of and .

(ii) In [9], S-S Byun extends the previous results to Orlicz spaces. He recalls the following definitions.

Definition 6. A positive function defined on is called a Young function if it is increasing, convex and satisfies

Definition 7. One says that the Young function if it satisfies the following conditions: for some numbers .

Definition 8. Let be a Young function. The Orlicz space    is the linear space of all measurable functions satisfying Equipped with the norm is a Banach space.

The following result is proved in [9]: let be a Young function. There exist a small and a positive constant such that, if the nonlinearity satisfies (11), (12), (13), and (8), if satisfies (9), and if , then the unique weak solution to (10) satisfies , with the estimate

Remark 9. If and inequality (19) reduces to only (14).

4. (Functions of) Bounded Variation

The notion of functions of bounded variation is closely related to the notion of measure and the following usual definitions are useful.

4.1. Measures

Definition 10. Let be an open subset of , and [resp., ] the space of continuous functions [resp., ] with compact support in .(i)A measure on is a linear functional that is continuous; in the following sense, for all compact in , there exists a constant such that for all whose support is contained in . We write . In [10], Schwartz introduces a topology on and is continuous on this topological space.(ii) is said to be bounded on if, in (21), the constant is independent of . The space of bounded measures on is denoted by .(iii)The conjugate of is given by (iv) is called real if , for all .(v) is called positive if , for all .

Definition 11 (absolute value of a measure). If is a complex or real measure, its absolute value, denoted by , is the map

Definition 12 (total variation of a positive bounded measure). Let be a positive, bounded measure on . Its total variation, denoted by or , is

4.2. Space BV or BV

Let be an interval in . The following alternative definitions are well known. A function is of bounded variation:(i)if it can be expressed in the form , where are nondecreasing bounded functions,(ii)if the interval is divided up by points then, there exists a constant , independent of the mode of division, such that and the upper bound of the sum (26) is called the total variation of on and is denoted by .  For these definitions, see Titchmarsh [11, page 355] and Riesz and Sz-Nagy [12, page 10].

We have also the following proposition: let . Then defines a distribution . In Schwartz [10, page 53] it is proved that, in order for the derivative of to be a measure, it is necessary and sufficient that is of bounded variation on every finite interval.

4.3. Space BV or BV

Let be an open set in . The following definitions are equivalent (cf. Brezis [13, page 153]).

A function is of bounded variation(i)if all first derivatives of , in the distributional sense, that is, in , are bounded measures,(ii)if there exists a constant such that (iii)if there exists a constant such that for all open set and all with . Moreover, in (27) and (28) we can take . Here , and is the distributional gradient of .

We have also the following proposition: a function is of bounded variation if its first distributional derivatives are bounded measures. Then, the gradient is a bounded, vector-valued measure whose absolute value is the map where and is a bounded positive measure. The total variation of is (cf. F. Demengel and G. Demengel [14, page 303]).

4.4. Space BV

This section is already presented in [7]. Let be an open subset of with a smooth boundary. A function has a bounded variation, that is, , if , in the distributional sense, is a vector-valued Radon measure of finite total mass. Let  be a BV-seminorm. In Dávila [15], the following property of is proved: there exists a positive constant , which depends on , such that, for every family of nonnegative radial mollifiers satisfying we have

4.5. Application

In [16], B. Merlet shows, by means of the above property of , that, if , there exists a lifting of (i.e., , for all ) such that .

5. Discrete Energy

Let , be a set of points on the unit-sphere .

5.1. Discrete Energy of

Different discrete energies are associated with . Let , .

Definition 13. The Coulomb [resp., logarithmic] energy associated with is [resp., ].

Definition 14. More generally, the -energy associated with is

Remark 15. We have

Remark 16. is the energy of the points , on the surface of the sphere, interacting through a potential (cf. Rakhmanov et al. [17]).
The article [17] is devoted to extremal energy for : Bounds for , and explicit formula for points on that yields good estimates for are obtained. The authors point out important applications of the determination of to geometry, chemistry, physics, and crystallography and give references for the history of related researches.

5.2. A More General Discrete Energy-Like Function on the Unit Sphere

In [18], Cheviakov et al. introduce the following definition.

Definition 17. The discrete energy-like function associated with is

Application. In [18], is related to the mean first passage time (MFPT) for a Brownian particle in the unit ball in that contains small locally circular absorbing windows on its boundary . Set . The function is solution to the Dirichlet-Neumann problem: <