Review Article | Open Access
A Survey of Some Topics Related to Differential Operators
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.
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 .
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 ).
Application. In , 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 ), 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 ) 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, ). 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 .
Related spaces are presented in .
3.2. BMO Nonlinearity (cf. Byun and Wang )
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. ).
Applications. (i) Let be a bounded, open subset of , , , is a vector field measurable in for almost every and continuous in for each . In , 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 , 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 : 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
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.
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 , 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 . 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 , 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
In , 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. ).
The article  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 , Cheviakov et al. introduce the following definition.
Definition 17. The discrete energy-like function associated with is
Application. In , 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: where is a diffusivity coefficient. The authors obtain three-term asymptotic expansions for , for the average MFPT , and for the principal eigenvalue of the Laplacian associated with the boundary conditions (38), when the area , as , and , . For instance, when the windows have common radius , they obtain Moreover, is minimized and the corresponding is maximized at the configuration that minimizes . The optimum arrangements that minimize are numerically computed by different methods.
6. Dissipative Operators
I restrict myself to definitions (cf. Pazy ). Let be a Banach space whose norm is denoted by , its dual, and . The value of at is denoted by .
Definition 18. For , we define the duality set by
Definition 19. A linear operator , in , with domain , is dissipative if, for every , there exists such that or, equivalently, if for all and . A dissipative operator is called -dissipative if, for all , the operator is surjective.
The Lumer-Phillips Theorem states that a densely defined operator in is the generator of a -semigroup (i.e., ) of contractions if and only if it is -dissipative (cf. ). I recall that the domain of the infinitesimal generator of is and, for any ,
7. Dynamical Systems
In this section, I restrict myself to several definitions and examples of dynamical systems and also definitions and some properties of limit and invariant sets. More investigations into this large topic are in preparation.
Different definitions of dynamical systems are given in the literature.
7.1.1. General Definitions
In (2011), Kloeden and Rasmussen give, in , the general definitions:
let [resp., ], , that is, [resp., ] and a metric space.
Definition 20. A dynamical system is a continuous map which satisfies the initial value condition and the group property
When the dynamical system is called continuous and when the dynamical system is called discrete.
Definition 21. A semidynamical system is a continuous map which satisfies the initial value condition and the semigroup property
When [resp., ] the semidynamical system is called continuous [resp., discrete].
7.1.2. “Classical” Dynamical Systems
Let be a Banach space. In 1969, Hale  gives the following definition.
Definition 22. A classical dynamical system on is a continuous function which satisfies for all and .
7.1.3. “Abstract” Dynamical System
In 1991, Haraux  defines abstract dynamical systems in a complete metric space in the following way.
Definition 24. An abstract dynamical system in is a family of mappings on which satisfy the following properties:(i)for each , is continuous,(ii),(iii),(iv)for each , is continuous.
7.1.4. Dafermos Definition
In [23, Definition 4.1], Dafermos gives the following definition in a metric space .
Definition 26. A Dafermos dynamical system in is a map which satisfies the initial condition the semigroup property and the continuity condition in , for all .
Remark 27. In the previous definitions the space [resp. , ] is called the phase space, and the set is called the positive orbit or trajectory through [resp., , ].
7.2.1. Continuous (Semi-)Dynamical Systems
Example 28 (cf. [21, Example 1]). Let be a continuous function and . Consider the autonomous differential equation with the initial condition If there exists a unique solution of (51) and (52) which depends continuously upon , then the mapping is a continuous semidynamical system on (Definitions 21 and 22).
Example 29 (cf. [20, Example 1.6]). Let be the Banach space of bounded continuous functions , under the usual norm . Let and a continuous function and suppose that there exists a unique solution of the autonomous delay differential equation with the condition Then is a continuous semidynamical system on .
Example 30 (see cf. [22, page 12]). Let be a real Banach space, a -dissipative operator with dense domain in (cf. Section 5), the semigroup of contractions generated by , and a Lipschitzian function on the bounded sets of . Let be the unique maximal solution of Set . Let such that for all and , for all , . For , is solution of (57) for . Set and denote by the distance induced on by the norm in . Then is an abstract dynamical system (Definition 24) on .
Example 31. Dafermos dynamical systems are adapted to equations of linear viscoelasticity. See Section 10.
Example 32 (see cf. ). Let be a metric space and be a continuous function. By iteration, , for all , , is well defined. The mapping is a discrete semidynamical system (Definition 21). Now, suppose is an homeomorphism; that is, it is continuous and invertible with continuous inverse. Then, the mapping defined by (58) can be extended to by and is a discrete dynamical system.
7.2.2. Discrete-Time Dynamical Systems and Fractals
What is known as a discrete time dynamical system in Pesin and Climenhaga  is related to a map from a set to itself. Let be any set and a map . By iteration, is well defined for every and we have the semigroup property: for any integer .
Definition 33. Let . The sequence of points , is called the trajectory or orbit of .
7.2.3. Examples with Chaotic Behavior
Example 34 (the Cantor set). Consider, in , the intervals , , and and the map defined by on and on . The range of is and does not lie in its domain of definition. The domain of definition of is , that is, the union of the intervals: obtained by removing the open middle third of each interval and and so on. The domain on which every iterate is defined is exactly the Cantor set (cf. Figure 1).
Example 35 (the Sierpinski triangle (or gasket)). An equilateral triangle is divided into four smaller triangles, each similar to the first and congruent to each other. Then, the middle triangle is removed, and the iterating procedure is iterated on the remaining three, and so on. The fractal obtained as the limit of this procedure is the Sierpinki triangle (cf. Figure 2). With this procedure is associated the following algorithm. If , and only in this case, it is possible to define a function such that for every (cf. Schroeder  which calls this procedure “Sir Pinski game”).
7.3. Limit and Invariant Sets
Definition 36 (see cf. ). Let be a dynamical system in a metric space . The -limit set [resp., -limit set] of a point is defined by In the case of a semidynamical system () the notion of a -limit set is not defined. In the same way, and -limit sets of a subspace are defined by (62) where is replaced by .
Remark 37. The and -limit sets may be defined as follows:
Remark 38. Similar definitions are valid for classical and abstract dynamical systems.
Definition 39. Let be a semidynamical system on the metric space (Definition 21). A subset is called invariant under if
It is called positively [resp., negatively] invariant if In the case of discrete-time dynamical system and of Example 32, invariance is equivalent to .
Properties of -Limit and Invariant Sets. Let be a metric space, a semidynamical system on , and . If the trajectory through is precompact, then has the following properties: If , does not need to be connected (cf. [20, Exercise 1.10]).
In the case of an abstract dynamical system , on a complete metric space, previous results hold with , for , if is relatively compact in (see [28, page 122]). But in a complete metric space a subset is relatively compact if and only if it is precompact (cf. ).
8. Equal-Area Condition
Equal-area type conditions appear, as sufficient or necessary conditions, in the formation of layers (internal or superficial) in stationary solutions to various singularly perturbed reaction-diffusion systems. In the recent works do Nascimento , Crema and do Nascimento , and do Nascimento and de Moura , the authors prove the necessity of suitable equal-area condition for the formation of internal or (and) superficial transition layers in this type of problems.
Example 40. A simple particular case of problems studied in  is the elliptic boundary value problem where is a smooth domain in , , such that there exist , with .
Let be a smooth -dimensional compact manifold without boundary. It is proved that, if (67) has a family of solutions which develop an internal transition layer with interface connecting the states to , then, necessarily, the simple equal-area condition is satisfied.
Example 41. In , the following stationary system is considered: where is a smooth domain in , , , , , and are sufficiently smooth -valued functions, and .
Let be an open connected set in , be an -dimensional compact connected orientable manifold whose boundary is such that is an -dimensional submanifold of . The authors give a definition of a family of internal transition layer solutions to (69) in with interface , depending on two functions , on . They show that, for such a family, there exists [resp., ] such that [resp., ] on compact sets of [resp., in ]. They also prove that, if a family of internal transition layer solutions to (69) exists, then on and necessarily the equal-area condition is satisfied. Several concrete applications of these results are presented in the paper.
Example 42. In , is a bounded domain in , , with boundary , is a -dimensional surface with a boundary which is assumed to be a -dimensional compact surface without boundary with , and intersects transversally. The authors define a family of solutions , , to the elliptic boundary value problem: which develops internal and superficial transition layers, depending on some smooth functions , , with interfaces and , respectively.
Here , , , and are of class and is the exterior normal vector field on . It is proved that the equal-area conditions are necessary for the existence of such solutions.
9. Inertial Manifolds
Let be a Hilbert space and the semigroup associated with an evolution equation of the form with the initial condition , where is a linear operator and a nonlinear one. When an inertial manifold exists for problem (73), the restriction of (73) to reduces to a finite dimensional ordinary differential equation (80), which is an exact copy of the initial system (cf. ). The manifold is usually viewed as the graph of a suitable smooth function , where is an orthogonal projection on with finite-dimensional range, and .
After some definitions, the following sections will be devoted to the existence of an inertial manifold for (73) and the presentation of results on the existence and the behavior of inertial manifolds for phase-field equations.
Let be a metric space and a continuous semigroup on .
Definition 43. A set is an inertial manifold for if(i) is a finite-dimensional Lipschitz manifold in ,(ii) is of class ,(iii) is positively invariant under the flow that is, , for all ,(iv) is exponentially attracting; that is, there exists a constant such that, for any , there exists a constant such that where is the Hausdorff semidistance (see Section 1.8 in Huet ); compare Bonfoh et al. [34, page 164].
Definition 44 (see Temam ). Let and an open subset of such that ; is said to be absorbing for in if, for any bounded set in , there exists such that , for all . In particular, if is bounded, , for .
Remark 45. Similar definitions were given in  and Luskin and Sell , in the case of Hilbert spaces. In  the smoothness of and the continuity of the semigroup are not parts of the definition of an inertial manifold.
In a Hilbert space whose norm is denoted by , consider problem (73) where is an unbounded linear operator in , strictly positive (i.e., such that for all in the domain of ), and self-adjoint. It is assumed that is compact. Thanks to assumptions on , it is possible to define its powers defined on , for all . When see also Huet , Kato , Schechter . For , we denote by the ball, in , with center of radius ; that is, It is assumed that satisfies the following properties.
. There exists such that is Lipschitz on the bounded sets of with values in and
. If problem (73) has a unique solution , for all , and the map is continuous from into itself, for all .
. The semigroup possesses an absorbing set such that(1), for all ,(2)the -limit set of is the maximal attractor for in .(3) is chosen such that is included in the ball of (cf. (75)).
9.3.1. The Prepared Equation
The prepared equation is equivalent to the original one for large. Let be a function such that The aim of the prepared equation is to avoid the difficulties related to the behavior of the nonlinear term for large values of . Let , , and The prepared equation associated with (73) is of the form where is chosen in . Let be the semigroup associated with (80). The following is assumed.
. The ball is absorbing for .
Thanks to the assumptions on , is a bounded operator and a Lipschitz mapping from into . In particular where is a constant which depends on .
Let , and defined in . Under the assumptions of , there exists an orthonormal basis in , where is the eigenvector of corresponding to the eigenvalue , with , as . Set where is the orthogonal projector, in , onto the space spanned by . The projections and commute with , for all .
Definition 46. Let be the space defined by where is a Lipschitz function with , that satisfies for all , .
Remark 47. is finite dimensional.
is a complete metric space for the distance
9.3.3. Construction of a Map
Let , . Thanks to assumptions on and , the problem has a unique solution . Then, the problem has a unique solution . Therefore (cf. Lemma 2.3, page 420 in ). The mapping is defined by with Let denote the right-hand side of (87). Then
Under assumptions , if it is possible to find , , and , such that is a strict contraction of into itself, that is, with , for all , then, has a fixed point , and the graph of is an inertial manifold for (73) or (80).
9.4. Inertial Manifolds for Phase-Field Equations
Example 48 (cf. Bonfoh ). Set , , and and denote by the operator . Consider the phase-field equations , in , with the boundary conditions and the initial conditions where , , and satisfies the conditions The limit problem () is given by (90) where : the boundary conditions (91), and the initial condition . Set Let denote the usual norm in and set , . The norms are equivalent to the usual Sobolev norms on . Several Banach spaces whose norms depend on are introduced. In particular Under additional assumptions on the regularity of the data, there exist semigroups , global attractors , and exponential attractors for problems , (cf. Chepyzhov and Vishik ). It is proved that and converge as , to some lifting of and , in (cf. (96)). Moreover, there exists , independent of , such that are bounded absorbing sets for in and in , respectively, where