WEAK FORMULATION OF SINGULAR DIFFERENTIAL EXPRESSIONS IN SPACES OF FUNCTIONS WITH MINIMAL DERIVATIVES

A weak formulation for singular symmetric di ﬀ erential expressions is presented in spaces of functions which possess minimal di ﬀ erentiability requirements. These spaces are used to characterize the domains of the various operators associated with such expressions. In particular, domains of self-adjoint di ﬀ erential operators are characterized


Introduction
Application of the general theory of self-adjoint operators to the spectral representation of operators associated with the formally self-adjoint differential expression u = 1 w n k=0 (−1) k p n−k u (k) (k) (1.1) was carried out to a completion by many researchers in this field. A complete account of this theory can be found in [1,11]. Account for the parallel theory of partial differential and difference operators can be found in [2,5]. On the other hand, the differential expression (1.1) gives rise to the formal sesquilinear form a(u,v) = n k=0 p n−k u (k) v (k) (1.2) encountered in the course of studying weak formulations of differential equations. Unlike the differential expressions, the theory behind the sesquilinear forms (1.2) is not yet fully developed. The most general treatment we have so far is for the case when such forms are semibounded or sectorial [10]. The classical Lax-Milgram theorem which is widely used in treatments involving the bilinear forms (1.2) assumes that the underlying form is positive and continuous. While such assumptions suffice to handle regular and some classes of singular differential expressions, they are not sufficient to handle the general singular expressions as they need not be semibounded. The importance of such a theory stems from the many important applications it would have in areas such as the calculus of variations and numerical solutions of differential equations. For some of these applications the reader is referred to the papers [3,4,7,9] and the references therein. In [6] a variational formulation of the second order differential expression was presented in regular as well as singular cases. Although no assumptions of semiboundedness were made there, the treatment has two drawbacks. In a general setting, the presentation depended on the existence of a maximal space of definition inferred from Zorn's lemma (see [6, page 43]). The difficulty with this space is the lack of a satisfactory concrete characterization to render it useful for further development. In a more special setting, the treatment relied on more concrete spaces but they require full differentiability assumptions and thus no use is made of the reduced order of differentiation granted by the variational setting ( [6, page 48]). This makes the presentation particularly unattractive if we want to devise Galerkin-like numerical methods to solve singular differential equations. These two drawbacks are eliminated in this work. We give here a weak formulation of the more general differential expression (1.1) in spaces which require differentiation properties dictated only by what is necessary for the sesquilinear form (1.2) to be meaningful. We also give full characterizations of various operators associated with the formal operator in terms of these spaces. These characterizations include the most interesting operators associated with , namely, self-adjoint operators. This paper is organized as follows. After this introduction we give a preliminary section in which the notation and the results frequently used in this work are given. The weak formulation of the problem is done in Section 3. In this section the working spaces are defined, the variational form of the problem is set and its equivalence to the original problem is established. In Section 4 some further properties of the defined spaces are explored.

Preliminaries
The following notation will be used in this paper. D(a, b) denotes the space of test functions on the interval (a,b), −∞ ≤ a < b ≤ ∞, and L(a, b) its dual with respect to the following topology. Denoting by ·, · the pairing between D(a, b) and L(a, b), a functional f ∈ L(a, b) if and only if for each compact interval [α,β] there is a constant C and an integer r ≥ 0 such that for every function v ∈ D(a, b) with support in [α,β] (C and r generally dependent on [α,β]). L 2 w (a,b) denotes the Hilbert space of complex-valued square integrable functions on the interval (a,b) with respect to the almost everywhere positive weight w. The inner product and norm in this space are denoted by ·, · w and · w , respectively. AC (k) (a,b) denotes the space of functions that are absolutely continuous on any compact subinterval of (a,b) together with their derivatives up to order k inclusive. AC(a,b) is used in place of AC (0) (a,b). L 1 loc (a,b) denotes the space of functions which are integrable on every finite sub-interval [α,β] of (a,b). The kth classical derivative of a function u will be denoted as usual by u (k) whereas the notation u [k] will be used to denote the kth pseudo-derivative of u defined by the formulae (see also [11]). Consider the formally self-adjoint differential expression defined on the interval (a,b), where w > 0 almost everywhere on (a,b), the coefficient functions p 0 , p 1 , ..., p n are real valued and 1/ p 0 , p 1 ,..., p n ,w ∈ L 1 loc (a,b). If a, b are finite and the functions 1/ p 0 , p 1 , ..., p n , w are integrable on (a,b) then this expression is said to be regular, otherwise it is singular.
The expression defines the following operators in L 2 w (a,b): (1) The "maximal" operator L whose domain Ᏸ is given by (2) The operator L 0 whose domain Ᏸ 0 is given by Ᏸ 0 = u ∈ Ᏸ : u has compact support in (a,b) , (2.5) (3) The "minimal" operator L 0 whose domain Ᏸ 0 is given by Note that (see [11]) [u,v](a) and [u,v](b) both exist for all u,v ∈ Ᏸ. All three operators are densely defined and the following relationships hold among them 694 Weak formulation for singular differential operators where denotes operator closure. In particular, the operators L 0 , L 0 are symmetric and the operators L 0 , L are closed. For λ ∈ C, Im(λ) = 0, put ᏺ λ = Ker(L − λI). Since the operator has real coefficients, u ∈ D if and only if u ∈ D and Lu = λu if and only if Lu = λu. The common dimension d of the spaces ᏺ λ and ᏺ λ is called the deficiency index of the operator L 0 . In fact, 0 ≤ d ≤ 2n and is independent of λ as long as Im(λ) = 0. Now for a fixed λ ∈ C\R, the subspaces Ᏸ 0 , ᏺ λ and ᏺ λ are linearly independent (see [8,11]) and For any u ∈ Ᏸ write where u 0 ∈ Ᏸ 0 , u λ ∈ ᏺ λ and u λ ∈ ᏺ λ . Then Formula (2.9) shows that L 0 is self-adjoint if and only if d = 0. Various characterizations of the domains Ᏸ of self-adjoint extensions L of the operator L 0 are given in [11] and elsewhere. We state here two characterizations which will be used in this work.
Theorem 2.1. Any self-adjoint extension L of the operator L 0 is characterized by a unitary transformation U : ᏺ λ → ᏺ λ such that (2.12) In other words, there is a one to one correspondence between self-adjoint extensions of L 0 and unitary transformations from ᏺ λ to ᏺ λ . In what follows we summarize some results from [6] which will also be needed in this work. From now on, when we state that a complex number exists or is defined we also mean that it is finite. For functions u,v ∈ AC (n−1) (a,b), we introduce the formal M. A. El-Gebeily 695 sesquilinear form if the integral exists. Let us also introduce the brackets and note that In a similar fashion to the Lagrange expressions we Obviously, if all parts of the above equation exist, then if and only if {u, v} b a = 0. For convenience, the following theorem is reproduced from [6]. Theorem 2.3. For every u ∈ Ᏸ 0 and v ∈ Ᏸ, a(u,v) exists and (2.20) It immediately follows from (2.18) that for all u ∈ Ᏸ 0 and v ∈ Ᏸ. Hence the description (2.6) of the domain of the minimal operator Ᏸ 0 may be sharpened to

Weak formulation
Note that the first and last expressions in (2.18) require 2n pseudo derivatives to be formed whereas the middle expression requires only n derivatives. We are thus led to considering the problem of obtaining a weak formulation for the expression in spaces that require only n derivatives. In this section we give such a formulation within the framework of the space L 2 w (a,b). As stated in the introduction, no assumptions are being made about the semiboundedness of the operators or the forms involved.
Define the following dense subspaces of L 2 w (a,b): Some comments on the choice of the above spaces are now in order. The choice of the space ᐂ was mainly motivated by the requirement that Ᏸ 0 ⊂ ᐂ. This requirement, together with the general assumptions we made about the coefficient functions, grant only the local integrability of the derivatives of the functions in ᐂ. The space ᐆ is so chosen to include the space Ᏸ whose functions have 2n − 1 absolutely continuous pseudoderivatives on the interval (a,b). Consequently, for a function u ∈ Ᏸ, u [n] = p 0 u (n) ∈ AC(a,b). From this one could infer a local L p property for any p, 1 ≤ p ≤ ∞. The choice of L ∞ loc (a,b) is forced by the natural duality with the properties of the space ᐂ in order to insure the existence of the integrals b a u [n] v (n) . Finally the space ᐆ 0 is chosen to include Ᏸ 0 and, at the same time not to exceed the differentiability properties granted by functions in the space ᐆ. It will be shown below that these spaces are dense in L 2 w (a,b) and give rise to a satisfactory theory for the weak formulation of the singular differentiable operators.
One is interested, in general, in solving variational equations of the form where f ∈ L 2 w (a,b) and v varies in some convenient space ᐃ. The equality (3.2) means that a continuity requirement with respect to the norm · w has to be imposed on the form a(u,·) over ᐃ. As we will see, this continuity requirement plays a crucial role in recovering the domains of definition of the operators associated with . Since this is the M. A. El-Gebeily 697 only continuity property we are going to need, the phrase "with respect to norm · w " will be dropped from this point on.
Proof. For u ∈ ᐆ 0 and v ∈ Ᏸ, u,Lv w exists and, from the definition of ᐆ 0 , {v, u} b a = 0, hence (see the Preliminaries) a(u,v) is defined and the result follows from (2.16).
Next we will show that Ᏸ is precisely the subspace of ᐆ for which the continuity property of the previous corollary holds. Before establishing this we need the following property.
Proof. Let u ∈ Ᏸ 0 . Clearly u satisfies the two properties defining the space ᐆ. On the other hand, let p 0 u (n) = g. (3.12)

M. A. El-Gebeily 699
Then g is absolutely continuous on the support of u. Furthermore, therefore the local integrability of 1/ p 0 implies the integrability of u (n) . Thus, u ∈ ᐂ.
Proof. Denote the right-hand side of the above equation by Ᏸ 1 . For u ∈ Ᏸ 1 define the antilinear functional G u (·) on ᐂ by (3.14) Then G u (·) is continuous on ᐂ. Since ᐂ is dense in L 2 w (a,b) we can extend G u (·) to all of L 2 w (a,b). Hence, by the Riesz representation theorem, there is a unique element Now notice that Ᏸ ⊂ Ᏸ 1 and for u ∈ Ᏸ we have This means that the operator T is densely defined and agrees with L on Ᏸ. That is, L ⊂ T. It follows that T * ⊂ L * = L 0 . Therefore T * is a symmetric closed operator. For v ∈ Ᏸ 0 with supp(v) = [α,β], u ∈ Ᏸ 1 we have 700 Weak formulation for singular differential operators In analogy with this result, we have the following theorem. Proof.
(1) This part is an immediate consequence of Theorems 3.3 and 3.6, and the den- that is, a(u,·) is continuous on ᐂ. Hence, u ∈ Ᏹ 0 . On the other hand, if u ∈ Ᏹ 0 , then a(u,·) is continuous on ᐂ and can be extended by continuity to all of L 2 w (a,b). In particular a(u,·) is continuous on Ᏸ and, by Lemma 3.2, Hence, the mapping v → u,Lv w is continuous on Ᏸ. Therefore, u ∈ D(L * ) = D(L 0 ) = Ᏸ 0 .
We next give a characterization of self-adjoint extensions of L 0 in terms of unitary operators between the spaces ᏺ λ and ᏺ λ and the space ᐆ 0 . The following theorem may be regarded as a counterpart of Theorem 2.1. Proof. Denote the right-hand side of (3.25) by Ᏸ 1 . It is straightforward to check that (3.26) The continuity of a(u,·) and (λU + λI)u λ ,· w on ᐂ imply the continuity of a(u 0 ,·) on ᐂ. Since u 0 ∈ ᐆ 0 , we get, by the second part of Theorem 3.7, that u 0 ∈ Ᏸ 0 . Hence, u ∈ Ᏸ.
The converse statement follows from the characterization in Theorem 2.1 and the first part of this theorem since the definition of Ᏸ implies that

Further properties and characterizations
In this section, we give further properties and alternative characterizations of the weak spaces ᐆ, ᐆ 0 and the domains of self-adjoint extensions of L 0 in terms of the so called "boundary condition functions." It was shown in the previous section that a(·,·) is defined on ᐆ × ᐂ. Since Ᏸ 0 ⊂ ᐂ, then a(·,·) is defined on ᐆ × Ᏸ 0 and, for a fixed u ∈ Ᏸ 0 , the mapping v → a(v,u) is continuous on ᐆ. The question is, how far can we push the space Ᏸ 0 and retain continuity on ᐆ? The answer is in the corollary to the following lemma. Proof. The proof is similar to that Theorem 2.3 with Ᏸ replaced by ᐆ.
Corollary 4.2. For every u ∈ Ᏸ 0 , the mapping v → a(u,v) is continuous on ᐆ.
We also have the following weakened definition of the space ᐆ 0 . for all functions ϕ ∈ ᏺ λ + ᏺ λ .
We turn now to characterizations of domains of self-adjoint extensions of L 0 that parallel Theorem 2.2. It was shown in [11] that the domain of definition Ᏸ of self adjoint extensions L of L 0 are characterized by functions w 1 ,w 2 ,...,w d ∈ Ᏸ satisfying conditions 1, 2 of Theorem 2.2 such that Ᏸ = Ᏸ 0 span w 1 ,w 2 ,...,w d . (4.8) Define the space with v 0 ∈ Ᏸ 0 . Using Lemma 4.1 we get Furthermore, since u, Lv w exists, it follows that a(u,v) exists. Equation (4.10) now follows from (2.16).
The foregoing theorem tells us that domains of the type (4.9) cannot be hoped to characterize all self-adjoint extensions of L 0 . They rather characterize extensions for which the boundary condition functions satisfy {w i ,w j } b a = 0, i, j = 1,2,...,d. This class of extensions will be called Class I. The following converse theorem applies to this class.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
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