A Survey of Some Topics Related to Differential Operators

ThispaperistheresultofinvestigationssuggestedbyrecentpublicationsandcompletestheworkofHuet,2010.Thetopics,whicharedealtwith,concernsomespacesoffunctionsandpropertiesofsolutionsoflinearandnonlinear,stationaryandevolutiondifferentialequations,namely,existence,spectralproperties,resonances,singularperturbations,boundarylayers,andinertialmanifolds.Theyarepresentedinthealphabeticalorder.TheaimofthisdocumentandofHuet,2010,istobeausefulreferencefor(young)researchersinmathematicsandappliedsciences.


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].
An asymptotic expansion of the bottom virtual eigenvalue   () of   , as  < 0 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  ↑ 0, for the nonbottom virtual eigenvalues of   , { k (), k ∈ Z  + ; 0 < |k| ≤ }, where  = −((+1)/2) if  is odd and  =  − (/2) if  is even.If  is odd, Φ(− 2 ) is a meromorphic operator function, and the leading terms of the asymptotic estimates of  k () 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.

BMO and Related Spaces
Definition 2. Let sup  be the supremum over all cubes  ⊂ R  with edges parallel to the coordinate axes, ℓ() the sidelength of , and   the mean value of  over .The John-Nirenberg's BMO = BMO(R  ) space (cf.John and Nirenberg [4]) is the space of locally integrable complex-valued functions  defined on R  , such that         BMO = (sup The space of functions of bounded mean oscillation, modulo constants, equipped with the above norm, is a Banach space. Definition 3 (real Hardy space  1 (R  )).A function  ∈  1 (R  ) if and only if where where the inf is taken on all decompositions of  of the form (4),  1 (R  ) is a Banach space.In Fefferman and Stein [6,Theorem 2], it is proved that the dual of  1 (R  ) is BMO.For definitions and properties of Hardy-spaces   (R  ), see [6].Related spaces are presented in [7].
(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 [0, ∞) is called a Young function if it is increasing, convex and satisfies Definition 7. One says that the Young function  ∈ Δ 2 ∪ ∇ 2 if it satisfies the following conditions: for some numbers ,   > 1.
Definition 8. Let  ∈ Δ 2 ∪ ∇ 2 be a Young function.The Orlicz space   (Ω) is the linear space of all measurable functions Equipped with the norm (Ω) is a Banach space.

(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.(i) A measure  on Ω is a linear functional that is continuous; in the following sense, for all  compact in Ω, there exists a constant   such that     ⟨, ⟩     ≤           ∞ (21) for all  ∈ C  (Ω, C) whose support is contained in .We write  ∈ M(Ω).In [10], Schwartz introduces a topology on C  (Ω, C) and  is continuous on this topological space.(ii)  is said to be bounded on Ω if, in (21), the constant is independent of .The space of bounded measures on Ω is denoted by M 1 (Ω).(iii) The conjugate  of  is given by ⟨, ⟩ = ⟨, ⟩. ( 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 Ω.
then, there exists a constant  > 0, 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 (, ).
We have also the following proposition: let  ∈  1 loc (R).Then  defines a distribution  =   ∈ D  (R).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.A function  ∈ (iii) if there exists a constant  such that for all open set  ⊂ Ω and all ℎ ∈ R  with |ℎ| < dist(, ∁Ω).Moreover, in (27) and (28) we can take  = ||∇||  1 (Ω) .Here  ℎ () = ( + ℎ), and ∇ is the distributional gradient of .
We have also the following proposition: a function  ∈  1 (Ω) 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 ⟨∇, ⟩ = ∑ (cf.F. Demengel and G. Demengel [14, page 303]).
Let Ω be an open subset of R  with a smooth boundary.A function  ∈  1 loc (Ω, R  ) has a bounded variation, that is,  ∈ BV(Ω, R  ), if ∇, in the distributional sense, is a vectorvalued Radon measure of finite total mass.Let |f| BV = ∫ Ω |∇f| be a BV-seminorm.In Dávila [15], the following property of || BV is proved: there exists a positive constant , which depends on , such that, for every family of nonnegative radial mollifiers
The article [17] is devoted to extremal energy for   : Bounds for E(, ), −2 <  < 2 and explicit formula for  points on  2 that yields good estimates for E(, ) are obtained.The authors point out important applications of the determination of E(, ) to geometry, chemistry, physics, and crystallography and give references for the history of related researches.

A More General
Application.In [18], H(  ) is related to the mean first passage time (MFPT) V() for a Brownian particle in the unit ball Ω in R 3 that contains  small locally circular absorbing windows Ω   on its boundary Ω =  2 .Set Ω  = ⋃  =1 Ω   .The function V() is solution to the Dirichlet-Neumann problem: where  is a diffusivity coefficient.The authors obtain threeterm asymptotic expansions for V(), for the average MFPT V = (1/|Ω|) ∫ Ω V(), and for the principal eigenvalue  of the Laplacian associated with the boundary conditions (38), when the area For instance, when the windows have common radius  ≪ 1, they obtain Moreover, V is minimized and the corresponding  is maximized at the configuration   that minimizes H(  ).The optimum arrangements { 1 , . . .,   } that minimize H(  ) are numerically computed by different methods.
The Lumer-Phillips Theorem states that a densely defined operator  in  is the generator of a   -semigroup () (i.e., lim  → 0 () = ) of contractions if and only if it is dissipative (cf.[19]).I recall that the domain () of the infinitesimal generator  of () is and, for any  ∈ (),

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.

Definitions
7.1.1.General Definitions.In (2011), Kloeden and Rasmussen give, in [20], the general definitions: let , N] and a metric space .Definition 20.A dynamical system is a continuous map Φ : T ×  →  which satisfies the initial value condition Φ (0, ) =  ∀ ∈  (43) and the group property When T = R the dynamical system is called continuous and when T = Z the dynamical system is called discrete.Definition 21.A semidynamical system is a continuous map Φ : T +  ×  →  which satisfies the initial value condition Φ (0, ) =  ∀ ∈  (45) and the semigroup property When

"Classical" Dynamical Systems.
Let  be a Banach space.In 1969, Hale [21] gives the following definition.Definition 22.A classical dynamical system on  is a continuous function  : R + ×  →  which satisfies for all ,  ≥ 0 and  ∈ .
Remark 23.In Definition 22, the mapping  is a continuous semidynamical system in terms of Definition 21.

"Abstract" Dynamical System.
In 1991, Haraux [22] defines abstract dynamical systems in a complete metric space (, ) in the following way.
the semigroup property and the continuity condition in , for all  ∈ R + .
Remark 27.In the previous definitions the space  [resp., ] is called the phase space, and the set with the initial condition If there exists a unique solution (, ) of ( 51) and ( 52) which depends continuously upon , , then the mapping with the condition Then is a continuous semidynamical system on .Example 32 (see cf. [20]).Let  be a metric space and  :  →  be a continuous function.By iteration,  +1 () = (  )() =   (), 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 Φ defined by ( 58) can be extended to Z − by and Φ is a discrete dynamical system.

Discrete-Time Dynamical Systems and Fractals.
What is known as a discrete time dynamical system in Pesin and Climenhaga [24] is related to a map from a set to itself.Let  be any set and  a map  :  → .By iteration,  +1 () = (  ()) = ( ∘   )() is well defined for every  ∈ N and we have the semigroup property: for any integer ,  ≥ 0.

Examples with Chaotic Behavior
Example 34 (the Cantor set).Consider, in R, the intervals  obtained by removing the open middle third of each interval  1 and  2 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  ∈  → () ∈ R 2 such that   () ∈  for every  (cf.Schroeder [25] which calls this procedure "Sir Pinski game").
The Cantor set and the Sierpinski triangle are examples of fractals (cf.[26,27]).

Limit and Invariant Sets
Definition 36 (see cf. [20]).Let Φ : T ×  →  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 (Φ :  +  × → ) 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: In the case of discrete-time dynamical system and of Example 32, invariance is equivalent to () = .
In the case of an abstract dynamical system {(()} ≥0 , on a complete metric space (, ), previous results hold with Φ(, ) = (), 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.[29]).

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 [30], Crema and do Nascimento [31], and do Nascimento and de Moura [32], 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 [30] is the elliptic boundary value problem where Ω is a smooth domain in R  ,  ≥ 1,  : R → R such that there exist , ,  > , with () = () = 0.
Let Γ ⊂ Ω be a smooth ( − 1)-dimensional compact manifold without boundary.It is proved that, if (67) has a family {  } of solutions which develop an internal transition layer with interface Γ connecting the states  to , then, necessarily, the simple equal-area condition is satisfied.
Here  ∈  1 (Ω),  > 0,  : Ω×R → R, and  : Ω×R → R are of class  1 and n is the exterior normal vector field on Ω.It is proved that the equal-area conditions

Inertial Manifolds
9.1.Setup.Let  be a Hilbert space and () the semigroup associated with an evolution equation of the form with the initial condition (0) =   ∈ , 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.[33]).The manifold  is usually viewed as the graph of a suitable smooth function Φ :  → , 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.9.2.Definitions.Let (, ) be a metric space and () a continuous semigroup on .Remark 45.Similar definitions were given in [33] and Luskin and Sell [35], in the case of Hilbert spaces.In [33] the smoothness of  and the continuity of the semigroup () are not parts of the definition of an inertial manifold.

The Prepared Equation.
The prepared equation is equivalent to the original one for  large.Let  : R + → [0, 1] 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   () = (/),  > 0, and The prepared equation associated with (73) is of the form where  is chosen in H 3 (3).Let   () be the semigroup associated with (80).The following is assumed.
where   is a constant which depends on .9.3.2.Space F   .Let  ∈ N, ,  > 0 and  defined in  1 .Under the assumptions of , there exists an orthonormal basis   in , where   is the eigenvector of  corresponding to the eigenvalue   , with where   is the orthogonal projector, in , onto the space spanned by  1 , . . .  .The projections  and  commute with   , for all  ∈ R.

Mathieu-Hill Type Equations
where  and  are constants, where  is any smooth periodic function of period 1 with mean 0 (see Coddington and Levinson [41] and Wang and Guo [42]).Physical problems leading to Mathieu or Hill equations often require solutions with periodicity, called oscillatory solutions.Therefore, to find conditions on the data for which the above equations have a fundamental system of periodic solutions is a central problem.

Its Equation.
In [43], A. R. Its considers the Schrödinger equation on the positive semiaxis where  is a smooth periodic function, with period 1 and mean 0,  is a real number, and the parameters ,  satisfy the relations  −  ≥ −1 and 2 −  > 0 (cf. Figure 3).He proves that (104) has oscillatory solutions when  >  − 1.If  =  − 1, the solutions are oscillatory or not.In all cases, asymptotic formulas for the solutions are stated, as  → ∞.His method is based on a transformation which leads to a Hill-type equation and Floquet functions. Let be the eigenvalues of a suitable matrix which depends on , , , and ], and Λ the diagonal matrix diag{ 1 ,  2 }.The matrix Λ is a crucial tool in the diagonalizable method used by the authors.When  =  − 1, three cases are studied separately as  1 −  2 belongs to N, or as it does not belong to N 0 or as it is equal to 0. In each case, very sharp asymptotic formulas are obtained for a fundamental system of solutions to (104), as  → ∞.In particular, when  1 −  2 ∈ N  , a logarithmic term appears in the formulas.For instance, when  1 −  2 ∈ N the following formulas are obtained: where  1,2  are scalar valued, bounded, continuous, periodic functions with period 1 which are recursively calculated and  ̸ = 0 is a suitable number.This logarithmic term is missing in Its formulas.

(Equations with) Memory
Equations with memory are differential equations which are influenced by the past of one or more variables.In this section, I restrict myself to a few examples.I present the equation with memory studied by Dafermos [45] in the context of linear viscoelasticity and examples of singularly perturbed equations with memory.
The function ℎ() is said to be of order .

Nodes and Nodal
where  is a bounded domain in R  .The nodal set   is defined as The nodal domains of   are the connected components of \ N(  ).The authors construct a domain  ⊂ R 2 on which the second eigenvalue has a nodal set disjoint from the boundary, whereas the nodal line conjecture, first mentioned by Payne [52], stated that Example 55.In [53], Bartsch and Weth consider the nonlinear elliptic Dirichlet problem: where Ω is a bounded domain in R  ,  ≥ 2,  > 0,  is a small positive parameter, and  grows superlinearly and subcritically.They study the number of nodal solutions, that is, sign-changing solutions of (129) and their nodal domains.They show that the number of nodal solutions can be expressed as a Lyusternik-Schnirelman category (cf.Lyusternik and Schnirelman [54]), of a suitable inclusion between two spaces which involve the shape of Ω.

Resonances
Only a few aspects of this large topic are presented here.More aspects are in preparation.

The Case of a Vibrating String
13.1.1.Free Motion (cf.Schwartz [56]).Consider a homogeneous vibrating string, with linear density , subjected to a constant tension .The position of the string, fixed at the end points  = 0,  = , is given by solutions of the wave equation with the boundary conditions where V = √/ has the dimension of a velocity.To solve problem (131) and ( 132) the initial values of the position and of the velocity of the string are needed; that is, are called the eigenvalues and sin(/) the eigenfunctions of the problem.The constants   ,   are the Fourier coefficients of suitable extensions of   and  1 to periodic functions.

The Case of a Forced Motion: Resonance (cf. Courant
and Hilbert [50]).We suppose that the vibrating string is under the influence of an external force which has the Fourier expansion: Then, the deflection of the string is solution to with the initial conditions (133).We look for a solution of the form The coefficients   () must be solutions of the equations The general solution of the homogeneous equation associated with (139) is A particular solution of (139) is obtained by the method of variation of constants (cf.Coddington [57]).Finally, where the constants   ,   are determined by the initial conditions (133).Now, suppose Q  () =  cos + sin .Then, if  ̸ = / , (, ), in (138), is a linear combination of a sinusoidal function of frequency  and one of frequencies /.But, if  = /,   () contains terms of the forms  sin  and  cos  which are unbounded, we say that resonance occurs.) .

Resonances as Poles of Complex Functions (cf. Zworski
( The solution to system (143) involves eigenvalues and eigenfunctions of ) . ( Then P  is an unbounded self-adjoint operator on H and its eigenvalues are real.Let (P  ) be the spectrum of P  .It is pointed out, in [58], that  ∈  (P  ) ⇐⇒  is a pole of (P  − ) −1 : H → H.
where  is defined in (142) and The operator P  is not self-adjoint on H.The eigenvalues and the eigenfunctions of P  are still defined by and the solutions of (149) are given by superpositions of solutions of the form  (, ) = exp (−)  () , but the eigenvalues  =  +  are not real any more.An elementary calculus shows that ( − ) =  2 , where  ∈ R is an eigenvalue of   ; that is,  = /2,  2 =  2 −( 2 /4).Here, again, the eigenvalues of P  are poles of (P  − ) −1 : H → H. Now, since  ∉ (P  ), for a given , the equation has a solution.But, if  is small enough,  =  + (/2) is a pole of (P  − ) −1 very close to , and, for a suitable choice of , the solution of (153) can be "enormous" (cf.[58]).We say that a resonance occurs.
where  > 0,  ∈  2 loc (R) is periodic with period 1,  is a real parameter indexing the equations of the family, and  > 0 is such that 2/ is irrational.Let or as it completely covers a spectral band of   or it contains exactly one edge or two edges (Figure 4) of a spectral band.
In each case, under specific additional assumptions, the authors describe the nature of the spectrum of (154) and state asymptotic formulas, as  → 0.
In [60], they are interested in the spectrum of (154), in intervals  such that, for all  ∈ , F() covers the edges of two neighboring spectral bands of   and the spectral gap located between them (Figure 4).Let Γ  be the real isoenergy curve associated with (154) and  0 ,   the connected components of Γ  in a periodicity cell.To each of these loops, one associates a sequence of energies in ,    ,  = {0, }, and, near each    , an exponentially small interval    such that the spectrum of (154), in , is contained in the union of these intervals.The location and the nature of the spectrum of (154) are investigated in the union   0 ∪    , in the resonant case, that is, when   0 and    intersect with each other.
and g are sufficiently smooth R  -valued functions, and k∇k = ( 1 ∇V 1 , . . .,   ∇V  ).Let U be an open connected set in Ω, Γ ⊂ U be an (−1)dimensional compact connected orientable manifold whose boundary Γ is such that Γ ∩ Ω is an ( − 2)-dimensional submanifold of Ω.The authors give a definition of a family of internal transition layer solutions {(  , k  )}, 0 <  <   } to (69) in U with interface Γ, depending on two functions ,  ∈   (U), () < () on Γ.They show that, for such a family, there exists   [resp., k  ] such that   →   [resp., k  → k  ] on compact sets of U\Γ [resp., in U].They also prove that, if a family of internal transition layer solutions to (69) exists, then (,   (), k  ()) = 0 on U \ Γ and necessarily the equal-area condition ∫ , , k  ()) )  = 0 (70) is satisfied.Several concrete applications of these results are presented in the paper.
the existence of such solutions.

10. 1 .
Mathieu and Hill Equations.The real Mathieu [resp., Hill] equation has the form
each function   is supported on a ball   and has integral zero, and sup ∈  |  ()| ≤ 1/|  |.
31.Dafermos dynamical systems are adapted to equations of linear viscoelasticity.See Section 10.