DOCTORA L T H E S I S
Department of Engineering Sciences and Mathematics Division of Mathematical Sciences
Maximal Regularity of the Solutions
for some Degenerate Differential
Equations and their Applications
Raya Akhmetkaliyeva
ISSN 1402-1544 ISBN 978-91-7790-100-6 (print)
ISBN 978-91-7790-101-3 (pdf) Luleå University of Technology 2018
Ra
ya
Akhmetkaliy
ev
a Maximal Regular
ity of the Solutions for some Degenerate Differ
ential Equations and their
Applications
Maximal Regularity of the Solutions
for some Degenerate Differential
Equations and their Applications
Raya Akhmetkaliyeva
Department of Engineering Sciences and Mathematics Division of Mathematical Sciences
Luleå University of Technology SE-971 87 Luleå, Sweden &
Department of Fundamental Mathematics Faculty of Mechanics and Mathematics L.N. Gumilyov Eurasian National University
Printed by Luleå University of Technology, Graphic Production 2018 ISSN 1402-1544 ISBN 978-91-7790-100-6 (print) ISBN 978-91-7790-101-3 (pdf) Luleå 2018 www.ltu.se
Key words and phrases. Hilbert space, differential equations, differen-tial operators, existence, uniqueness, estimates of norms of the solutions, separation, coercive estimates, approximation numbers, smoothness of the solutions, resolvent, completely continuous resolvent, compactness, radius of fredholmness.
Abstract
This PhD thesis deals with the study of existence and uniqueness together with coercive estimates for solutions of certain differential equations.
The thesis consists of six papers (papers A, B, C, D, E and F), two appendices and an introduction, which put these papers and appendices into a more general frame and which also serves as an overview of this interesting field of mathematics.
In the text below the functions r = r(x), q = q(x), m = m(x) etc. are functions on (−∞, +∞), which are different but well defined in each paper. Paper A deals with the study of separation and approximation properties for the differential operator
ly = −y00+ r(x)y0+ s(x)¯y0
in the Hilbert space L2 := L2(R), R = (−∞, +∞), (here ¯y is the complex
conjugate of y). A coercive estimate for the solution of the second order dif-ferential equation ly = f is obtained and its applications to spectral problems for the corresponding differential operator l is demonstrated. Some sufficient conditions for the existence of the solutions of a class of nonlinear second order differential equations on the real axis are obtained.
In paper B necessary and sufficient conditions for the compactness of the
resolvent of the second order degenerate differential operator l in L2 is
ob-tained. We also discuss the two-sided estimates for the radius of fredholmness of this operator.
In paper C we consider the minimal closed differential operator
Ly = −ρ(x)(ρ(x)y0)0+ r(x)y0+ q(x)y
in L2(R), where ρ = ρ(x), r = r(x) are continuously differentiable functions,
and q = q(x) is a continuous function. In this paper we show that the operator L is continuously invertible when these coefficients satisfy some suitable conditions and obtain the following estimate for y ∈ D(L):
k − ρ(ρy0)0k
2+ kry0k2+ kqyk2 ≤ ckLyk2,
iv
where D(L) is the domain of L.
In papers D, E, and F various differential equations of the third order of the form
−m1(x) m2(x) (m3(x)y0) 00
+ [q(x) + ir(x) + λ]y = f (x) (0.1)
are studied in the space Lp(R).
In paper D we investigate the case when m1 = m3 = m and m2 = 1.
Moreover, in paper E the equation (0.1) is studied when m3= 1. Finally, in
paper F the equation (0.1) is investigated under certain additional conditions on mj(x) (j = 1, 2, 3).
For these equations we establish sufficient conditions for the existence and uniqueness of the solution, and also prove an estimate of the form
m1(x)(m2(x) (m3(x)y0)0) 0 p p+ k(q(x) + ir(x) + λ)yk p p≤ c kf (x)k p p
Preface
This PhD thesis consists of six papers (papers A, B, C, D, E and F), two appendices and an introduction, which puts these papers and appendices into a more general frame.
[A] K. N. Ospanov and R. D. Akhmetkaliyeva, Separation and the exis-tence theorem for second order nonlinear differential equation, Electron. J. Qual. Theory Differ. Equ., (2012), no. 66, 12 pp.
[A1] R. D. Akhmetkaliyeva, Appendix to Paper A.
[B] R. D. Akhmetkaliyeva, K. N. Ospanov and A. Zulkhazhav, Compact-ness conditions and estimates for the Fredholm radius of the resolvent of the degenerate second order differential operator, AIP Conference Proceedings, 1637 (2014), no. 13, 13-17, doi: 10.1063/1.4904559.
[C] K. N. Ospanov and R. D. Akhmetkaliyeva, Some inequalities for sec-ond order differential operators with unbounded drift, Eurasian Math. J., 6 (2015), no.2, 63-74.
[D] R. D. Akhmetkaliyeva, K. N. Ospanov, L. -E. Persson and P. Wall, Some new results concerning a class of third-order differential equations, Appl. Anal. 94 (2015), no. 2, 419-434.
[E] R. D. Akhmetkaliyeva, On solvability of third-order singular differen-tial equation, Functional analysis in interdisciplinary applications, Springer Proc. Math. Stat., 216 (2017), 106-112.
[E1] R. D. Akhmetkaliyeva, Appendix to Paper E.
[F]∗ R. D. Akhmetkaliyeva, On maximal regularity of singular third-order
differential equations, Lule˚a University of Technology, Department of
vi
ematical Sciences, Research Report 1 (2018), 15 pp, submitted. * An abbreviated version of this paper is also published as:
[*] R. D. Akhmetkaliyeva, Coercive solvability of the differential equation of the third order with complex-valued coefficients, Vestnik ENU, 95 (2013), no. 4, 355–361 [In Russian].
Acknowledgment
I want to express my deep gratitude to my main supervisors Professor
Lars-Erik Persson (Department of Engineering Sciences and Mathematics, Lule˚a
University of Technology, Sweden) and Professor Kordan Ospanov (Eurasian National University, Kazakhstan) for their valuable remarks and attention to my work and their constant support.
I also want to thank my co-supervisor Professor Peter Wall (Department
of Engineering Sciences and Mathematics, Lule˚a University of Technology,
Sweden) for his help in editing my work and for constant support. I also thank Professors Mukhtarbay Otelbaev and Ryskul Oinarov (Eurasian Na-tional University, Kazakhstan) for generously sharing their valuable knowl-edge in the field with me and for their constant support during my studies.
Furthermore, I thank everyone at the Department of Mathematics at Lule˚a
University of Technology for their friendly attitude to me and for the inspir-ing atmosphere.
I thank Lule˚a University of Technology and L. N. Gumilyov Eurasian
Na-tional University for giving me an opportunity to participate in their part-nership program in research and postgraduate education in mathematics. I also thank both universities for financial support which made my studies possible.
Finally, I am grateful to my family and friends for helping me in every possible way.
Introduction
This PhD thesis deals with the smoothness and approximation properties of solutions of differential equations defined in the Lebesgue space and having real and sometimes complex coefficients.
The main questions in the investigation of differential equations can be classified into the following three categories: existence, uniqueness and qual-itative behavior of the solutions. The first two questions are responsible for the accordance of the equations as a mathematical model of the real pro-cess, and the third question is necessary to investigate in order to know more about the nature of the process. In the study of the qualitative behavior of solutions of linear and nonlinear differential equations we are interested in the following questions:
1) the smoothness of the solutions;
2) estimates of solutions in different weighted norms; 3) approximation properties of the solutions.
The problem of smoothness for solutions of elliptic equations and esti-mates of solutions in various norms are well studied in the case when the domain is bounded and the coefficients are reasonable ”regular”. In this case we have methods which are nowadays developed to the classical perfec-tion and presented in detail in well-known monographs. A fairly complete bibliography of works in this field can be found e.g. in the books of O. A. Ladyzhenskaya and N. N. Ural’tseva [58], J.-L. Lions and E. Magenes [59].
Unfortunately, these methods are not applicable for differential equations given in an unbounded domain and with increasing (not integrable) coeffi-cients. Studies of problems of this type were first made by W. N. Everitt and M. Giertz [39-44] as singular Sturm-Liouville problems. In particular, the formulation of the fundamental problem of separability for a differential operator belong to them. Moreover, in [39-44] the same authors basically elucidated the conditions on the potential function q(x), providing the sepa-rability of the Sturm-Liouville operator
Ly(x) = −y00(x) + q(x)y(x), x ∈ R.
x
In these papers W. N. Everitt and M. Giertz called the indicated operator
separable in the space L2 = L2(−∞, +∞), if from y ∈ D(L), Ly ∈ L2 it
follows that q(x)y, y00∈ L2.
Here and in the sequel c (c1, c2, etc) denotes a positive constant which
may be different in various places.
It is well-known that the separability of the operator L is equivalent to the existence of the estimate
ky00kL 2(R)+ kqykL2(R)≤ c kLykL 2(R)+ kykL2(R) , y ∈ D(L), (0.2)
where D(L) is the domain of L. In [39-44] it was shown that if inf q(x) > −∞ and
q−14(x)
00
q14(x) ∈ L1, then the operator L is separable. Moreover,
an example of a separable operator L with non-smooth potential q was given. In the paper [45] an example of a non-separable operator L with an infinitely differentiable but rapidly oscillating potential was given. Inde-pendently of each other F. V. Atkinson [13], K. H. Boimatov [23], [26], M. Otelbaev [88] and D. Z. Raimbekov [102] weakened the condition used by W. N. Everitt and M. Giertz. In particular, in [23], [88] and [102], the con-dition
q−14(x)
00
q14(x) ∈ L1 was replaced by weaker conditions (different
by different authors), which are similar to the known conditions of Levitan - Titchmarsh, which commonly are used in investigations of the resolvent (concerning these conditions, see e.g. [25], [87] and [109]). In [88] the
prob-lem of separability was considered not only in the Hilbert space L2, but also
in non-Hilbert weighted spaces Lp,l(where l is a continuous weight function).
Here Lp,l is defined by the norm
kf kp,l := +∞ Z −∞ |f (x)l(x)|pdx 1 p (1 ≤ p < +∞).
In particular, it was shown that the separability of the Sturm-Liouville oper-ator holds for an extensive class of rapidly oscillating potentials (for example,
q(x) = e|x|sin2e|x|5). Later on M. Otelbaev proposed a special method with
local representation of the resolvent to solve the problem concerning the smoothness of solutions of some differential equations, which he called varia-tional. Multivariate equations were considered in [24], where K. H. Boimatov essentially verbatim transferred results from [23] to a class of elliptic oper-ators. The connection of separation with some concrete physical problems was noted in [83].
The existence and smoothness of solutions of nonlinear differential equa-tions (with a singular potential) for unbounded domains equipped with the
Sturm-Liouville equation was considered by M. B. Muratbekov and M. Otel-baev [80]. Later on this problem was investigated in the works T. T. Amanova [9] and M. B. Muratbekov [78].
In [51] the authors investigated the separability of the nonlinear Sturm-Liouville operator
Ly = −y00+ q(x, y)y
in the space L1(−∞, +∞). Moreover, in [1], [17], [29], [74], [114] and [115]
the differential expression
Ly = − (P (x)y0)0+ Q(x)y, x ∈ (−∞, +∞),
with operator coefficients was considered.
We have thus motivated the fact that in the case when the differential equation is given in an infinite domain and has unbounded coefficients, the problem of determining the estimates of separability of the type (0.2) for the corresponding differential operator is meaningful. The presence of estimates of separability allows us to accurately describe the class of functions, where the generalized solution of the singular boundary value problem for the dif-ferential equation belongs. At the same time the estimate of separability provides a precise description of the domain generated by the indicated sin-gular boundary value problem for the differential operator. This domain is usually a weighted Sobolev space. Thus, if we have estimates of separability, then we can use the modern theory of function spaces to study qualitative properties of the solutions of singular differential equations. We recall that the famous scientist I. M. Gelfand considered that finding estimates of sepa-rability (maximal regularity) is one of the most central problems in the study of elliptic equations in the general theory of linear operators (see e.g. preface of the book [58, p.8]).
Separability of a wide class of linear elliptic differential operators was investigated in [11], [13], [23-44], [65-102] and [113-115], where, in particu-lar, important smoothness and approximation properties of solutions of these equations and the spectral properties of the associated singular differential and integral operators were investigated. The methods of proofs in the in-dicated works are based on deep facts of the theory of embedding between function spaces, of spectral theory of operators, and also widely used ad-vances in the theory of integral operators in function spaces and non-local apriori estimates of generalized solutions. These studies had an enormous influence on the development of the theory of singular differential equations, spectral theory of operators, the theory of weighted function spaces and in-tegral operators in them.
xii
Note that the results in all of these papers was concerned only with second order linear differential operators whose first order terms can be estimated in norms with the other terms involved. However, many practical problems lead us to study elliptic equations, whose properties depend strongly on the behavior of the components with intermediate derivatives of the solution in-volved and where we have no such norm estimate. Such equations are called in the literature degenerate differential equations. These include for
exam-ple equations of Schr¨odinger type with intermediate members with unlimited
potential from below, the Korteweg - de Vries type equation, where the coef-ficient of the junior term depends on the derivative of the unknown function, as well as the differential equation of oscillations in environments with resis-tance which is proportional to the velocity or acceleration (see [108]). Despite of this, the study of degenerate differential equations was carried out only in the symmetric case for the corresponding differential operators in [47], [54], [55] and [106], where, in particular, the problem of self-adjoint operators as-sessing their eigenvalues and determination of the structure of the spectrum was solved.
In stochastic analysis and in the theory of stochastic equations, the so-called generalized Ornstein-Uhlenbeck equation
Lu = −∆u + ∇u · b + cu = f (x) (0.3)
is widely used, where x ∈ Rn, b = b(x) is a continuously differentiable
vector-valued function, c = c(x) is a continuous function, f (x) ∈ L2(Rn) and the
point between the vectors denotes the scalar product in Rn, see e.g. [12],
[22], [34], [36], [46], [60], [61], [63], [66], [71], [98] and [100]. For example, in [98] the differential operator L corresponding to (0.3) acts as the generator of the semigroup of the transition of the stochastic process, which determines the n -dimensional Brownian motion with a single covariant matrix.
A solution u = u(x) from L2(Rn) of the equation (0.3) is said to be regular
if u ∈ W2
2(Rn). Further, if the solution u exists and satisfies the following
estimate
k∆uk2+ k∇u · bk2+ kcuk2≤ c (kf k2+ kuk2) ,
then we say that equation (0.3) is coercive solvable.
The function b = b(x) in (0.3) is called displacement or drift. When
b = 0, the equation (0.3) is the stationary Schr¨odinger equation, which has
been systematically studied for a long time in connection with quantum-mechanical applications (see [53], [55], [81] and [103]). If the coefficient b is different from zero and is not limited, then equation (0.3) is fundamentally
obtained by adding to the Schr¨odinger operator an operator with respect to small perturbations.
The study of equation (0.3) is interesting not only from the theoretical point of view. It originally appeared in the fundamental paper [84] in con-nection with the description of the Brownian motion of particles. Studies of such scientists as M. Smoluchowski, A. Fokker, M. Plank, H.C. Burger, R. Furth, L. Zernike, S. Goudsmit, M.C. Wang and others were devoted to this issue in the early 20th century. An overview of their results can be found in [112]. Along with (0.3) we investigate a more general equation:
Au = −∇ (Q∇u) + ∇u · b + cu = f (x), (0.4)
where Q = Q(x) is a real n × n - matrix. The differential expression Lu is a part of the Fokker-Planck and Cramer equations. In recent years the regularity of the solutions of equations (0.3) and (0.4) has been studied in [37], [38], [57], [68], [69] and [70]. Detailed information on other works can be found in the recently published monograph [22], which contains an ex-tensive bibliography. The problems of the propagation of small oscillations in viscoelastic compressible media [105], [111], the dynamics of a stratified compressible fluid [48], the motion of particles in media with a resistance proportional to the velocity [108], as well as, biology [52] and financial math-ematics [50] also lead to elliptic equations with displacement (0.3) and (0.4). The growth of the modulus of a vector (displacement) at infinity and its re-lation to the growth of other coefficients affects the solvability and regularity of equations (0.3) and (0.4). To compensate for the growth of b, the authors of [18], [49], [62], [67] and [99] regard both the solution and the right-hand side of these equations as elements of some weighted space whose weight is in some sense comparable with b. P. Cannarsa and V. Vespri [34] (see also [33]) introduced a weight V with the same asymptotics as the real part of c
and assumed that |b|/√V and c/V are bounded. They established the
exis-tence and uniqueness of the solution of (0.4) in the weighted Sobolev space H1
V(Rn).
A smaller amount of work is devoted to the nonweighted case. Moreover, A. Lunardi and V. Vespri [64] considered the case when c is bounded, and the displacement b has linear growth, and they proved the unique solvability
of (0.3) with f ∈ L2(Rn) in the space H1(Rn). Using this result, G.
Meta-fune [65] gave a characterization of the spectrum of the Ornstein-Uhlenbeck operator. Under certain conditions on bounded Q, b and c, P.J. Rabier [101]
proved the unique solvability of (0.4) in H2
(Rn). We also mention the paper
of M. Sobajima [107], where in the case of linear and logarithmic growth, respectively, b and c, the m - accretiveness and the m - sectoriality of the
xiv
operator A in (0.4) were shown. Similar results were established in [108]. We pronounce that in all these papers and in [70], [71] and [98] there remains to investigate the case when the displacement b in (0.4) have a stronger growth and oscillation, so that it can not be controlled by diffusion Q and potential c.
Moreover, the following result [77] of A.M. Molchanov for the
Sturm-Liouville operator is known: The resolvent L−1 of the operator
Ly = −y00+ q(x)y, q ≥ 1, x ∈ R,
is compact in the space L2if and only if lim
|x|→∞ x+d
R
x−d
q(t)dt = +∞ holds for each d > 0. The compactness of resolvents for a wide class of semibounded elliptic operators for which there exists an extension in the sense of Friedrichs were obtained in [21] and [94]. Such results for some non-self-adjoint operators are proved by applying the known results obtained for semibounded operators whose properties are close to self-adjoint operators.
In papers A, B and C of this PhD thesis (= [85], [3] and [86], respectively) we study the more general case concerning a degenerate differential equation having non-symmetric form. In the same papers we consider the question of solvability (apparently for the first time) for a quasilinear degenerate differ-ential equation. And we also investigate the spectral properties of the second order degenerate differential operators with complex coefficients. The nature of such operators is not close to semi-bounded operators. Hence, invertibility of degenerate differential operators, the compactness of their inverse opera-tor, and other questions about the structure of the spectrum have not been investigated so far. We considered an one-dimensional degenerate differential operator of the form
ly := −y00+ r(x)y0+ s(x)¯y0, x ∈ R,
with increasing ”intermediate” coefficients. Here, the free term is equal to
zero and the members rdxd and s dxd do not depend on the operator l1y ≡
−y00, in other words, they are not infinitesimal perturbations of the operator
l1 in some sense.
The problem about the structure of the spectrum for the degenerate op-erator l, as we saw above, has important practical applications. The fact that the operator l is not symmetric together with other problems raises an issue about the invertibility of the operator l.
A number ρA = inf T ∈σ∞(L2) kA − T kL 2→L2 −1 ,
is called the radius of fredholmness of a bounded operator A in L2 [92]. Here
σ∞(L2) denotes the set of all compact operators in L2.
In paper B we derived two-sided estimates of ρl−1 of the inverse operator
of the above operator l.
Some criteria of compactness of the resolvent for semi-bounded operators are discussed in [21], [77] and [95]. Estimates of the radius of fredholmness of embedding operators of Sobolev spaces are established in [92].
The approaches developed in the above studies also allows us to study some classes of non semibounded differential operators, i.e., such energy
spaces that are not enclosed in a Sobolev space. The non semibounded
operators include all differential operators of odd order. Linear and nonlin-ear differential operators of odd order were investigated e.g. in [7], [8], [10], [14-16], [19], [20], [79], [104] and [110]. However, all of them except those of Zh. Zh. Aytkozha and M. B. Muratbekov [15], A. Birgebaev and M. Otel-baev [20] and M. B. Muratbekov, M. M. Muratbekov and K. N. Ospanov [79] were devoted to the case of a real potential and in [15] and [20] the case of a Hilbert space was considered. Odd order differential equations with singular complex coefficients in Banach space have not been studied systematically. Such equations constantly arise in the application of the projection methods, in particular in Fourier’s method of separation of variables for solving partial differential equations.
In Papers D, E and F (=[4], [5] and [6], respectively) we investigate some more general third order equations than those above. Usually, the previous mentioned authors only consider equations of the type
Ly = −y000+ q(x)y = f (x),
where f = f (x) ∈ Lp(R), R = (−∞, +∞). However, we consider the more
general case, when the coefficients are not constant in the leading term. Before starting presentation of the results obtained in papers A, B, C, D, E and F we present a number of well-known necessary
Definitions, notations and auxiliary results
Rn is a n-dimensional real Euclidean space; in particular when n = 2
we obtain a two-dimensional Euclidean space of points z = (x, y), where −∞ < x < ∞, −∞ < y < ∞.
Ω denotes an open domain in Rn and by Ω we denote the closure of Ω.
Let α = (α1, α2, ..., αn), where αj ≥ 0 (j = 1, 2, ...n) are integers. We also
use the notation |α| = α1+ α2+ ... + αn.
xvi
partial derivatives of order up to l inclusive in Ω, which can be written as
Dα(u) := ∂ |α|u ∂xα1 1 ∂x α2 2 ...∂xαnn , where |α| ≤ l.
C∞(Ω) denotes the space of infinitely differentiable functions in Ω.
Definition 0.1. The set {x ∈ Ω : u(x) 6= 0} is called the support of the
function u = u(x) defined on the set Ω and it is denoted by supp u.
C0∞(Ω) denotes the set of infinitely differentiable and compactly
sup-ported functions in Ω.
L2 = L2(Ω) is the Hilbert space of Lebesgue measurable functions on Ω
with a finite norm
kuk2,Ω:= Z Ω |u|2dΩ 1 2 . Wk
2(Ω) denotes the space of functions from L2(Ω) having all the
general-ized Sobolev derivatives up to order k ≥ 1 also belonging to L2(Ω) with the
norm kukWk 2(Ω) := X |α|≤k Z Ω |Dαu|2dΩ 1 2 .
The domain of the operator A is denoted by D(A) and the range of A is denoted by R(A).
Definition 0.2. An operator A is called a bijection if, for any x1 and x2
belonging to D(A), such that Ax1= Ax2, it follows that x1 = x2.
If A maps D(A) onto R(A) bijectively, then there exists an inverse
mapping or inverse A−1 which maps R(A) onto D(A).
Definition 0.3. The operator A is said to be closed if, for every sequence
{xn} ⊂ D(A), the fact that xn → x0 and Axn → y0 implies that x0 ∈ D(A)
and y0= Ax0.
If the operator A is not closed, then sometimes it can be extended to be closed. This operation is called the closure of the operator A and the operator is called closable.
A criterion to guarantee that an operator has a closed extension: an
xn → 0 and Axn → y implies that y = 0.
Definition 0.4. An operator A is said to be completely continuous if it maps
every bounded set into a compact set or, for every bounded sequence {xn} of
elements of D(A), the sequence {Axn} contains a convergent subsequence.
Let X and Y be normed spaces and let A be a bounded operator from X to Y . We define a functional ϕ by
ϕ(x) = (x, ϕ) = (Ax, f ), x ∈ X, f ∈ Y∗, (0.5)
where Y∗ denotes the conjugate space of the space Y .
It is easy to see that ϕ is linear and D(ϕ) = X. Hence, according to
(0.5), for each f ∈ Y∗ there exists an element ϕ ∈ X∗, where X∗ is the
conjugate space to X. Thus a linear continuous operator ϕ = A∗f is given.
This operator A∗ is called the adjoint of A.
Definition 0.5. An operator A acting in the Hilbert space L2(Ω) is said to
be self-adjoint if it is symmetric, i.e., if the scalar product hAu, vi = hu, Avi for any u, v ∈ D(A) and from the identity
hAu, vi = hu, wi ,
where v and w are fixed, u is any element from D(A), it follows that v ∈ D(A) and w = Av.
Next we give the definition of Kolmogorov’s k-widths and their properties. Let M be a centrally symmetric subset of H (H is a Hilbert space), i.e., M = −M . The value dk = inf {Gk} sup u∈M inf v∈Gk ku − vk , k = 0, 1, 2, ...
is called Kolmogorov’s k-width of the set M , where Gk is a subset with
dimension k.
The k-widths dk (k = 1, 2, ...) have the following properties:
1) d0≤ d1≤ d2≤ ...;
2) dk( fM ) ≤ dk(M ), M ⊂ M, k = 1, 2, 3, ...;f
xviii
Let
◦
Ll
p(Ω, q) be the completion of C0∞(Ω), defined by the norm
(−∆) l 2u p Lp(Ω) + Z Ω q(t)|u(t)|pdt,
where q(t) is a nonnegative function, Ω is an open (bounded or unbounded)
set in Rn, l > 0, 1 ≤ p < ∞. We pronounce that the space
◦
Ll p(Ω, q)
uniquely arises in many situations in the study of differential equations.
We also define the following function q∗(x) introduced by M. Otelbaev
(see [97]): q∗(x) = inf Qd(x)⊆Ω d −1 : d−pl+n ≥ Z Qd(x) q(t)dt , (0.6)
where Qd(x) is a cube with sides equal to d and with center x ∈ Ω, pl > n.
Definition 0.6. Let B1 and B2 be Banach spaces. B1 is said to be embedded
in B2 if B1 is a subspace B2 and there is a constant c > 0 such that
kxkB
2 ≤ c kxkB1 for all x ∈ B1.
In this case we write B1,→ B2.
Definition 0.7. Let B1 and B2 be Banach spaces. Then a transformation
E mapping each element x from B1 to the same element in B2 is called the
embedding operator and denoted by E : B1 → B2.
We also need the following important result of M. Otelbayev [93]: Theorem 0.1. The embedding operator E :
◦
Llp(Ω, q) ,→ Lp is compact if and
only if
q∗(x) → ∞ when |x| → ∞.
Let B1 and B2 be Banach spaces and B1 ,→ B2.
Definition 0.8. The Kolmogorov k-width of the unit ball of the space B1 in
We introduce a function N (λ) = P
dk>λ
1 as the number of k-widths of the
embeddings B1,→ B2greater than λ > 0. N (λ) is also called the distribution
function of the k-widths dk.
We also observe that the k-widths dk can be recovered from their
distri-bution function using the formula
dk = inf{λ > 0 : N (λ) ≤ k}, for any k > 0.
Let N (λ) be a distribution function of the k-widths {dk} related to the
embedding
◦
Llp(Ω, q) ,→ Lp.
Then the following theorem by M. Otelbayev (see [89], [94]) holds: Theorem 0.2 Let pl > n. Then the following estimates
c−1λ−nlµ x ∈ Ω : q∗(x) ≤ λ−1l ≤ N (λ) ≤ cλ−n lµ x ∈ Ω : q∗(x) ≤ λ−1l
hold, where µ(·) is the Lebesgue measure and c depends only on p, l and n. It is easy to see that if d = 1 and the condition
sup |x−y|≤1 x,y∈Rn q(x) q(y) ≤ c (0.7) holds, then c−10 qpl−n(x) ≤ q∗(x) ≤ c 0qpl−n(x), where c0 > 1 and q∗(x) is
defined by (0.6). In this case Theorems 0.1 and 0.2 can be restated in terms of the function q(x) in the following way:
Theorem 0.3. Let pl > n and for a positive function q(x) the condition (0.7) holds. Then the embedding operator
◦
Ll
p(Ω, q) ,→ Lp is compact if and
only if
q(x) → ∞ when |x| → ∞.
Theorem 0.4. Let pl > n and the condition (0.7) holds. Then the following estimates c−1λ−nlµ x ∈ Ω : q(x) ≤ λ−l(pl−n)1 ≤ N (λ) ≤ cλ−nlµ x ∈ Ω : q(x) ≤ λ−l(pl−n)1
xx
hold, where µ(·) is the Lebesgue measure and c depends only on p, l and n. Theorem 0.5 (Schauder). Let D be a nonempty closed bounded convex subset of a Banach space X and let the operator A : X → X be compact and map D into itself. Then A has a fixed point in D.
Now we are ready to briefly present the most important results of the papers A, B, C, D, E and F. In the sequel the functions r(x), s(x), m(x) etc. are functions on (−∞, +∞), which are different but well defined in each paper.
In paper A we study a degenerate second order differential operator with complex coefficients.
Let l be the closure in L2 := L2(R), R = (−∞, +∞) of the expression
l0y = −y00+ r(x)y0+ s(x)¯y0
defined in the set C0∞(R) of all infinitely differentiable and compactly
sup-ported functions. Here r = r(x) and s = s(x) are complex-valued functions
and ¯y is the complex conjugate to y.
The operator l is said to be separable in L2if the following estimate holds:
ky00k2+ kry0k2+ ks¯y0k2≤ c (klyk2+ kyk2) , y ∈ D(l),
where k·k2 is the L2- norm. In this paper sufficient conditions for the
invert-ibility and separability of the differential operator l are obtained. Moreover,
spectral and approximate results for the inverse operator l−1 are achieved.
Moreover, by using a separation theorem, which is obtained for the linear case, the solvability of the degenerate nonlinear second order differential equation
−y00+ r(x, y)y0
= F (x ∈ R) (0.8)
is proved.
Definition 0.9. A function y ∈ L2 is called a solution of (0.8) if there is
a sequence of twice continuously differentiable functions {yn}∞n=1 such that
kθ(yn− y)k2 → 0, kθ(Lyn− f )k2→ 0 as n → ∞ for any θ ∈ C
∞ 0 (R). We denote αg,h(t) = kgkL2(0,t) h−1 L2(t,+∞) (t > 0), βg,h(τ ) = kgkL2(τ,0) h−1 L2(−∞,τ ) (τ < 0),
and γg,h = max sup t>0 αg,h(t), sup τ <0 βg,h(τ ) , where g and h are given functions.
By Cloc(1)(R) we denote the set of functions f such that ψf ∈ C(1)
(R) for all ψ ∈ C0∞(R).
The main results of this paper are the following:
Theorem 0.6. Let the functions r and s satisfy the conditions r, s ∈ Cloc(1)(R), Re r − |s| ≥ δ > 0, γ1,Re r< ∞.
Then l is invertible and l−1 is defined in all L
2.
Theorem 0.7. Assume that the functions r and s satisfy the conditions r, s ∈ Cloc(1)(R), Re r − ρ[|Im r| + |s|] ≥ δ > 0, γ1,Re r < ∞, 1 < ρ < 2, c−1 ≤ Re r(x)Re r(η) ≤ c at |x − η| ≤ 1, c > 1. (0.9)
Then, for y ∈ D(l) the estimate
ky00k2+ kry0k2+ ks¯y0k2 ≤ c klyk2 holds, i.e. the operator l is separable in L2.
Two crucial Lemmas (Lemmas 2.1 and 2.2) to prove these theorems were not proved in Paper A. However, a detailed proofs of these Lemmas
are included as Appendix [A1] of this PhD thesis. In particular, in the proof
of one of these Lemmas was used an important resuld from the theory of Hardy type inequalities (see [56]).
Theorem 0.8. Assume that the functions r and s satisfy (0.9) and let
lim
t→+∞α1,Re r(t) = 0, τ →−∞lim β1,Re r(τ ) = 0. Then l
−1 is completely continuous
in L2.
We assume that the conditions of Theorem 0.8 hold and consider the set
xxii Let dk = inf Σk⊂{Σk} sup y∈M inf w∈Σk ky − wk2 (k = 0, 1, 2, ...)
be the Kolmogorov’s widths of the set M in L2. Here {Σk} is a set of all
subspaces Σk of L2 whose dimensions are not greater than k. By N2(λ) we
denote the number of widths dk, which are not smaller than a given positive
number λ. In particular, estimates of the width’s distribution function
N2(λ) are important in some approximation problems of solutions of the
equation ly = f . In paper A also the following Theorems were stated and proved:
Theorem 0.9. Assume that the conditions of Theorem 0.8 are fulfilled and
let the function q = q(x) satisfy γq,Re r < ∞. Then the following estimates
hold:
c1λ−2µx : |q(x)| ≤ c−12 λ
−1 ≤ N
2(λ) ≤ c3λ−2µx : |q(x)| ≤ c2λ−1 ,
where µ is the Lebesgue measure.
Theorem 0.10. Let the function r be continuously differentiable with respect to both arguments and satisfy the following conditions
r ≥ δ0 √ 1 + x2 (δ 0> 0), sup x, η∈R: |x−y|≤1 sup A>0 sup
|C1|≤A,|C2|≤A,|C1−C2|≤A r(x,C1) r(η,C2) < ∞. Then there exists a solution y of (0.8), and
ky00k2+ k[r(·, y)]y0k2 < ∞.
In paper B, under conditions of separability, we obtained a necessary and
sufficient condition for the compactness of the operator l−1 in L2. Moreover,
we derived two-sided estimates of ρl−1 of the operator l−1. The main result
in paper B reads:
Theorem 0.11. Let the functions r = r(x) and s = s(x) be continuously differentiable and satisfy the conditions
|Re r| − ρ[|Im r| + |s|] ≥ δ > 0, γ1,Re r< ∞, ρ > 1,
Then, for the Fredholm radius ρl−1 of the operator l−1 the following esti-mates hold
c−14 ≤ ρl−1γ1, Re r ≤ c4.
In paper C we consider the minimal closed differential operator
Ly = −ρ(x)(ρ(x)y0)0+ r(x)y0+ q(x)y
in L2(R), where ρ = ρ(x) and r = r(x) are continuously differentiable
func-tions, and q = q(x) is a continuous function. We do not assume that ρ, r, q are bounded in R. In this paper we showed that the operator L is continu-ously invertible when these coefficients satisfy some suitable conditions and obtained the following estimate for y ∈ D(L)
k−ρ(ρy0)0k2+ kry0k2+ kqyk2≤ c kLyk2,
where D(L) is the domain of L, k · k2 is the norm in L2(R), and c is
independent of y.
Theorem 0.12. Let ρ = ρ(x) be a bounded continuously differentiable func-tion, and let r = r(x) and q = q(x) be continuous functions. Moreover, suppose that ρ ≥ 1, r and q satisfy the conditions
r ≥ ρ2, γ1,√r < ∞,
1 ≤ ρ(x) ≤ c 1 + x2N
for some N > 0,
and γq,r < ∞. Then L is continuously invertible and L−1 is defined on the
whole L2(R). Furthermore, there exists c such that
k−ρ(ρy0)0k2+ kry0k2+ kqyk2≤ c kLyk2, for any y ∈ D(L).
Papers D, E and F are related in the sense that the most general results are presented and proved in paper F. However, some basic results, ideas and techniques to derive these general results are developed already in papers D and E (c.f. also Remark 0.1). For example, in paper D we consider the third order differential equation with unbounded coefficients:
(L + λE) y := −m(x) (m(x)y0)00+ [q(x) + ir(x) + λ]y = f (x), (0.10)
where f ∈ Lp , λ ≥ 0, and where q(x), r(x) are continuous functions and
xxiv
In this paper D we study questions of the existence and uniqueness of the solutions of (0.10) and conditions, which for a solution y of (0.10) the following estimate holds:
km(x)(m(x)y0)00kpp+ k(q(x) + ir(x) + λ)ykpp≤ c kf (x)kpp. (0.11) We remark that when m(x) = 1 sufficient conditions for unique solvability of the equation (0.10) and the estimate of the form (0.11) for its solution in
the spaces Lp,lwere obtained by Zh. Zh. Aytkozha [14] and Zh. Zh. Aytkozha
and M. B. Muratbekov [15].
In the case when m(x) = 1 and r(x) = 0 the existence and uniqueness questions for the solutions of (0.10) and also non-local estimates of the solutions and its derivatives have also been studied in [7], [8] and [104].
Definition 0.10. A function y(x) ∈ Lp(R), is called a solution of (0.10),
if there is a sequence {yn}∞n=1 of continuously differentiable functions with
compact support, such that kyn− ykp → 0 and k(L + λE)yn− f kp → 0 as
n → ∞.
By C(k)
(R) (k = 1, 2, ...) we denote the set of all k times continuously differentiable functions ϕ(x) for which the value
k P j=0 sup x∈R |ϕ(j)(x)| is finite. Let Wλ(x) := |q(x) + λ + ir(x)| m2(x) .
Our main results in paper D read:
Theorem 0.13. Assume that the functions q = q(x) and r = r(x) are
continuous on R, m = m(x) ∈ Cloc(2)(R) and that the following conditions
hold: m(x) ≥ 1, q(x) m4(x) ≥ 1, r(x) ≥ 1, (0.12) c−1 ≤ m(x) m(η), q(x) q(η), r(x) r(η) ≤ c, x, η ∈ R, |x − η| ≤ 1, for some c > 0, (0.13) |m(j)(x)| ≤ c jm(x), x ∈ R, for some cj > 0, j = 1, 2, (0.14) and sup |x−η|≤1 |Wλ(x) − Wλ(η)| |Wλ(x)|ν|x − η|µ < +∞, 0 < ν < µ 3 + 1, µ ∈ (0, 1], λ ≥ 0. (0.15)
Then there exists a number λ0 ≥ 0, such that the equation (0.10) has a
solution y for all λ ≥ λ0.
Theorem 0.14. Let the functions q = q(x) and r = r(x) be continuous on
R, m = m(x) ∈ Cloc(3)(R) and satisfy the conditions (0.12) - (0.15) and
|m(3)(x)| ≤ c
3m(x), x ∈ R.
Then the solution of the equation (0.10) is unique and the estimate (0.11) holds.
In paper E we continue our investigations of this type by considering another similar case which required further developments of the methods. The obtained results were similar. So therefore we finish by describing the main results only in paper F. We only mention that a crucial Lemma (Lemma 4) in paper E was not proved there. However, in this PhD thesis a detailed
proof of this Lemma is included as Appendix [E1].
In paper F we investigate the more general problem of existence and uniqueness of solutions of the third order differential equations
−m1(x) m2(x) (m3(x)y0) 00
+ (q(x) + ir(x) + λ) y = f (x), (0.16)
where x ∈ R = (−∞, +∞), f ∈ Lp(R), 1 < p ≤ ∞ and λ ≥ 0. We
assume that q(x), r(x), m1(x) are continuous functions, and m2(x) ∈ C
(1) loc(R),
m3(x) ∈ C
(2)
loc(R). We also derive conditions so that for a solution y of (0.16)
the following estimate holds: m1(x) m2(x) (m3(x)y 0)00 p p+ k(q(x) + ir(x) + λ)yk p p≤ c kf (x)k p p. (0.17)
Our main results in Paper F are formulated in the following two Theorems:
Theorem 0.15. Assume that the functions q(x), r(x) and m1(x) are
contin-uous, m2(x) ∈ C
(1)
loc(R), m3(x) ∈ C (2)
loc(R) and satisfy the following conditions:
mj(x) ≥ 1 (j = 1, 2, 3), q(x) 3 Q k=1 m2 k(x) ≥ 1, r(x) ≥ 1, (0.18) c−1 ≤ mk(x) mk(η) ,q(x) q(η), r(x) r(η) ≤ c, (k = 1, 2, 3), x, η ∈ R, |x − η| ≤ 1, (0.19)
xxvi |m02(x)| ≤ cm2(x), |mj3(x)| ≤ cm3(x) (j = 1, 2), x ∈ R, (0.20) sup |x−η|≤1 |Wλ(x) − Wλ(η)| |Wλ(x)|ν|x − η|µ < +∞, 0 < ν < µ 3 + 1, µ ∈ (0, 1], λ ≥ 0. (0.21)
Then there exists a number λ0 ≥ 0, such that the equation (0.16) for all
λ ≥ λ0 has a solution y, where Wλ(x) :=
|q(x) + λ + ir(x)| 3 Q k=1 mk(x) .
Theorem 0.16. Let the functions q(x), r(x) be continuous, m1(x) ∈
Cloc(3)(R), m2(x) ∈ C (2)
loc(R), m3(x) ∈ C (2)
loc(R) and satisfy the conditions
(0.18)-(0.21) and (
|m(j)1 (x)| ≤ cm1(x), j = 1, 3,
|m(i)k (x)| ≤ cmk(x), k = 2, 3, i = 1, 2, x ∈ R.
Then the solution y of the equation (0.16) is unique and the estimate (0.17) holds.
By a solution we mean a solution in the sense of Definition 0.10 but with (0.10) replaced (0.16).
Remark 0.1. As previously mentioned some basic results, ideas, and
techniques, to prove these results were developed already in papers D and E. In particular, Theorem 0.15 coincides with Theorem 0.13 and Theorem 0.16
coincides with Theorem 0.14 for the case m2(x) ≡ 1 and m1(x) = m3(x).
Moreover, Theorem 0.15 coincides with Theorem 1 in paper E and Theorem
Bibliography
[1] A. A. Abudov, Separability of an operator generated by an
operator-differential expression, (Russian) Spectral theory of operators, ” `Elm”,
Baku, (1982), no. 4, 4-11.
[2] R. D. Akhmetkaliyeva, Coercive solvability of the differential equation of the third order with complex-valued coefficients, (Russian) Vestnik ENU, 95 (2013), no. 4, 355–361.
[3] R. D. Akhmetkaliyeva, K. N. Ospanov and A. Zulkhazhav, Compactness conditions and estimates for the Fredholm radius of the resolvent of the degenerate second order differential operator, AIP Conference Proceed-ings 1637, 13 (2014), 13-17, doi: 10.1063/1.4904559.
[4] R. D. Akhmetkaliyeva, K. N. Ospanov, L. -E. Persson and P. Wall, Some new results concerning a class of third order differential equations, Appl. Anal. 94 (2015), no. 2, 420-435.
[5] R. D. Akhmetkaliyeva, On solvability of third-order singular differential equation, Functional analysis in interdisciplinary applications, Springer Proc. Math. Stat., 216 (2017), 106-112.
[6] R. D. Akhmetkaliyeva, On maximal regularity of singular third-order
differential equations, Lule˚a University of Technology, Department of
Mathematical Sciences, Research Report 1 (2018), 15 pp, (submitted). [7] B. I. Aliev, A theorem of the separability of third-order ordinary
differ-ential equations on the half-line, (Russian) Spectral theory of operators
and its applications, ” `Elm”, Baku, (1989), no. 9, 3-10.
[8] T. T. Amanova, On the separability of a differential operator, (Russian) Izv.Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat., (1981), no. 3, 48-51.
xxviii BIBLIOGRAPHY [9] T. T. Amanova, Smoothness and approximation properties for solutions of two-term differential equations on an infinite interval, (Russian) The-sis for the degree of candidate of physical and mathematical sciences, Almaty, (1984).
[10] T. T. Amanova and M. B. Muratbekov, Smoothness of the solution of a nonlinear differential equation, (Russian. Kazakh summary) Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat., (1983), no. 5, 5-7.
[11] O. D. Apyshev and M. Otelbaev, The spectrum of a class of differential operators and some imbedding theorems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 739-764.
[12] W. Arendt, G. Metafune and D. Pallara, Schr¨odinger operators with
unbounded drift, J. Operator Theory 55 (2006), no. 1, 185-211.
[13] F. V. Atkinson, On some results of Everitt and Giertz, Proc. Roy. Soc. Edinburgh Sect A 71, part 2, (1973), 151-158.
[14] Zh. Zh. Aytkozha, On smoothness and approximation properties of solu-tions of differential equasolu-tions of odd order, (Russian) Thesis for the de-gree of candidate of physical and mathematical sciences, Almaty, (2003). [15] Zh. Zh. Aytkozha and M. B. Muratbekov, On smoothness and approxi-mation properties of solutions of nonlinear differential equations of third order with complex potential, (Russian) Abstracts of Republican Scien-tific Conference ”Theory of approximation and embedding of functional spaces”, Karaganda, (1991), 52.
[16] Zh. Zh. Aytkozha, M. B. Muratbekov and K. N. Ospanov, On the solv-ability of a class of nonlinear singular third-order equations, (Russian) Bulletin of Eurasian National University 46 (2005), no. 6, 10-15. [17] M. Bairamogly and A. A. Abudov, Essential selfadjointness of the
Sturm-Liouville operator with operator coefficients, (Russian) Spectral
theory of operators, ” `Elm”, Baku, (1982), no. 4, 12-20.
[18] A. Bensoussan and J.-L. Lions, Applications of variational inequalities in stochastic control, Studies in Mathematics and its Applications 12, North-Holland Publishing Co., Amsterdam-New York, (1982), 564 pp. [19] A. Birgebaev, Smooth solution of non-linear differential equation with
matrix potential, (Russian) The VII Scientific Conference of Mathemat-ics and MechanMathemat-ics, Almaty, (1989).
[20] A. Birgebaev and M. Otelbaev, Separability of a third-order nonlinear differential operator, (Russian) Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat., (1984), no. 3, 11-13.
[21] M. ˘S. Birman and B. S. Pavlov, On the complete continuity of certain
imbedding operators, Vestnik Leningrad. Univ 16 (1961), no. 1, 61-74.
[22] V. I. Bogachev, N. V. Krylov, M. R¨ockner and S. V. Shaposhnikov,
Fokker-Planck-Kolmogorov equations, Mathematical Surveys and Mono-graphs, 207 American Mathematical Society, Providence, (2015), 479 pp.
[23] K. H. Boimatov, Separability theorems for the Sturm-Liouville operator, (Russian) Mat. Zametki 14 (1973), 349-359; English translation in Math. Notes 14 (1974), 761-767.
[24] K. H. Boimatov, Theorems on the separation property, (Russian) Dokl. Akad. Nauk SSSR 213 (1973), 1009-1011; English translation in Soviet Math. Dokl. 14 (1973), 1826-1828.
[25] K. H. Boimatov, Asymptotic behavior of the spectrum of an operator differential equation, (Russian) Uspehi Mat. Nauk 28 (1973), no. 4(172), 207-208.
[26] K. H. Boimatov, The domain of definition of a Sturm-Liouville opera-tor, (Russian) Differencial’nye Uravnenija 12 (1976), no. 7, 1151-1160; English translation in Differential Equations 12 (1976), no. 7, 812-819. [27] K. Kh. Boimatov, Separability theorems, weighted spaces and their
ap-plications, (Russian) Trudy Math. Inst. Steklov 170 (1984), 37-76. [28] K. Kh. Boimatov, Coercive estimates and separability for second-order
elliptic differential equations, (Russian) Dokl. Akad. Nauk SSSR 301 (1988), no. 5, 1033-1036; English translation in Soviet Math. Dokl. 38 (1989), no. 1, 157-160.
[29] K. H. Boimatov and A. Sharifov, Coercive estimates and separability for differential operators of arbitrary order, (Russian) Uspekhi Mat. Nauk 44 (1989), no. 3(267), 147-148; English translation in Russian Math. Surveys 44 (1989), no. 3, 181-182.
[30] R. C. Brown, Separation and disconjugacy, JIPAM. J. Inequal. Pure Appl. Math. 4 (2003), no. 3, Article 56, 16 pp. (electronic).
xxx BIBLIOGRAPHY [31] R. C. Brown and D. B. Hinton, Two separation criteria for second order ordinary or partial differential operators, Math. Bohem. 124 (1999), no. 2-3, 273-292.
[32] R. C. Brown, D. B. Hinton and M. F. Shaw, Some separation criteria and inequalities associated with linear second order differential opera-tors, Function spaces and applications (Delhi, 1997), 7-35, Narosa, New Delhi, (2000).
[33] P. Cannarsa and V. Vespri, Generation of analytic semigroups by elliptic operators with unbounded coefficients, SIAM J. Math. Anal. 18 (1987), no. 3, 857-872.
[34] P. Cannarsa and V. Vespri, Generation of analytic semigroups in the Lp
topology by elliptic operators in Rn, Israel J. Math. 61 (1988), no. 3,
235-255.
[35] N. Chernyavskaya and L. Shuster, Weight summability of solutions of the Sturm-Liouville equation, J. Differential Equations 151 (1999), no. 2, 456-473.
[36] M. Chicco and M. Venturino, Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain, Ann. Mat. Pura Appl. (4) 178 (2000), 325-338.
[37] G. Cupini and S. Fornaro, Maximal regularity in Lp
(RN) for a class
of elliptic operators with unbounded coefficients, Differential Integral Equations 17 (2004), no. 3-4, 259-296.
[38] R. Donninger and B. Sch¨orkhuber, A spectral mapping theorem for
per-turbed Ornstein-Uhlenbeck operators on L2
(Rd), J. Funct. Anal. 268
(2015), no. 9, 2479-2524.
[39] W. N. Everitt and M. Giertz, Some properties of the domains of certain differential operators, Proc. London Math. Soc. (3) 23 (1971), 301-324. [40] W. N. Everitt and M. Giertz, Some inequalities associated with certain
ordinary differential operators, Math. Z. 126 (1972), 308-326.
[41] W. N. Everitt and M. Giertz, On some properties of the powers of a formally self-adjoint differential expression, Proc. London Math. Soc. (3) 24 (1972), 149-170.
[42] W. N. Everitt and M. Giertz, On some properties of the domains of powers of certain differential operators, Proc. London Math. Soc. (3) 24 (1972), 756-768.
[43] W. N. Everitt and M. Giertz, An example concerning the separation property of differential operators, Proc. Roy. Soc. Edinburgh Sect. A 71, part 2 (1973), 159-165.
[44] W. N. Everitt and M. Giertz, Inequalities and separation for Schr¨odinger
type operators in L2(Rn), Proc. Roy. Soc. Edinburgh Sect. A 79
(1977/78), no. 3-4, 257-265.
[45] W. N. Everitt, M. Giertz and J. Weidmann, Some remarks on a sep-aration and limit-point criterion of second-order, ordinary differential expressions, Math. Ann. 200, (1973), 335-346.
[46] S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Second order
elliptic operators in L2with first order degeneration at the boundary and
outward pointing drift, Commun. Pure Appl. Anal. 14 (2015), 407-419. [47] M. G. Gasymov, The distribution of eigenvalues of selfadjoint ordinary differential operators, (Russian) Dokl. Akad. Nauk SSSR 186 (1969), 753-756; English translation in Soviet Math. Dokl. 10 (1969), 646-650. [48] S.A. Gabov and A.G. Sveshnikov. Problems of stratified fluids dynamics,
Nauka, Moscow (1986).
[49] B. Goldys, On analyticity of Ornstein-Uhlenbeck semigroups, Atti Ac-cad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 (1999), no. 3, 131-140.
[50] F. Gozzi, R. Monte and V. Vespri, Generation of analytic semigroups and domain characterization for degenerate elliptic operators with un-bounded coefficients arising in financial mathematics. I. Differential In-tegral Equations 15, (2002), no.9, 1085-1128.
[51] `E. Z. Grinshpun and M. Otelbaev, Smoothness of solutions of a
nonlin-ear Sturm-Liouville equation in L1(−∞, +∞), Izv. Akad. Nauk Kazakh.
SSR Ser. Fiz.-Mat. (1984), no. 5, 26-29.
[52] D.-C. Jhwueng and V. Maroulas, Phylogenetic Ornstein-Uhlenbeck re-gression curves, Statist. Probab. Lett. 89 (2014), 110-117.
[53] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, (1995), 619 pp.
xxxii BIBLIOGRAPHY [54] A. G. Kostyuchenko, Some spectral properties of differential operators,
(Russian) Mat. Zh. 1 (1967), no. 3, 365-368.
[55] A. G. Kostyuchenko and I. S. Sargsyan, Distribution of eigenvalues, (Russian) Nauka, Moscow (1979), 400 pp.
[56] A. Kufner, L. E. Persson and N. Samko, Weighted Inequalities of Hardy-type, World Scientific Publishing, New Jersey (2017).
[57] P. C. Kunstmann and L. Weis, Maximal Lp -regularity for parabolic
equations, Fourier multiplier theorems and H∞- functional calculus,
Functional analytic methods for evolution equations. Lecture Notes in Math., Springer, Berlin, 1855 (2004), 65-311.
[58] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear El-liptic Equations, (Russian) Nauka, Moscow (1964).
[59] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Springer, Berlin (1972).
[60] L. Lorenzi and M. Bertoldi, Analytical methods for Markov semigroups, Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL,283 (2007), 526 pp.
[61] L. Lorenzi, A. Lunardi and R. Schnaubelt, Strong convergence of solu-tions to nonautonomous Kolmogorov equasolu-tions, Proc. Amer. Math. Soc. 144 (2016), no. 9, 3903-3917.
[62] A. Lunardi, On the Ornstein-Uhlenbeck operator in L2 spaces with
re-spect to invariant measures, Trans. Amer. Math. Soc. 349 (1997), no. 1, 155-169.
[63] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems
with unbounded coefficients in Rn, Studia Math., 128 (1998), no. 2,
171-198.
[64] A. Lunardi and V. Vespri, Generation of strongly continuous semigroups
by elliptic operators with unbounded coefficients in Lp
(Rn), Rend. Istit.
Mat. Univ. Trieste 28 (1996), suppl., 251-279 (1997).
[65] G. Metafune, Lp-spectrum of Ornstein-Uhlenbeck operators, Ann.
[66] G. Metafune and D. Pallara, Discreteness of the spectrum for some
dif-ferential operators with unbounded coefficients in Rn. Atti Accad. Naz.
Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat.Appl. 11, (2000), no. 1, 9-19.
[67] G. Metafune, D. Pallara and E. Priola, Spectrum of Ornstein-Uhlenbeck
operators in Lp spaces with respect to invariant measures, J. Funct.
Anal. 196 (2002), no. 1, 40-60.
[68] G. Metafune, D. Pallara, J. Pr¨uss and R. Schnaubelt, Lp -theory for
elliptic operators on Rdwith singular coefficients, Z. Anal. Anwendungen
24, (2005), no. 3, 497-521.
[69] G. Metafune, D. Pallara and V. Vespri, Lp-estimates for a class of elliptic
operators with unbounded coefficients in RN, Houston J. Math. 31,
(2005), no. 2, 605-620.
[70] G. Metafune, J. Pr¨uss, R. Schnaubelt and A, Rhandi, Lp -regularity for
elliptic operators with unbounded coefficients, Adv. Differential Equa-tions 10, (2005), no. 10, 1131-1164.
[71] G. Metafune, J. Pr¨uss, A, Rhandi and R. Schnaubelt, The domain of
the Ornstein-Uhlenbeck operator on an Lp-space with invariant measure,
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 2, 471-485.
[72] A. S. Mohamed, Separability of the Schr¨odinger operator using matrix
material [potential], Dokl. Akad. Nauk Respub. Tadzhikistan 35 (1992), no. 3, 156-159.
[73] A. S. Mohamed, Existence and uniqueness of the solution, separation for certain second order elliptic differential equation, Appl. Anal. 76 (2000), no. 3-4, 179-184.
[74] A. S. Mohamed and H. A. Atia, Separation of the Sturm-Liouville dif-ferential operator with an operator potential, Appl. Math. Comput. 156 (2004), no. 2, 387-394.
[75] A. S. Mohamed and H. A. Atia, Separation of the Schr¨odinger operator
with an operator potential in the Hilbert spaces, Appl. Anal. 84 (2005), no. 1, 103-110.
[76] A. S. Mohamed and H. A. Atia, Separation of the general second order elliptic differential operator with an operator potential in the weighted Hilbert spaces, Appl. Math. Comput. 162 (2005), no. 1, 155-163.
xxxiv BIBLIOGRAPHY
[77] A. M. Molˇchanov, On conditions for discreteness of the spectrum of
self-adjoint differential equations of the second order, Trudy Moskov. Mat. Obˇsˇc. 2 (1953), 169-199.
[78] M. B. Muratbekov, On the smoothness of solutions of degenerate
el-liptic equations and one-dimensional stationary nonlinear Schr¨odinger
equation, (Russian) Thesis for the degree of candidate of physical and mathematical sciences, Almaty, (1981).
[79] M. B. Muratbekov, M. M. Muratbekov and K. N. Ospanov, Coercive solvability of an odd-order differential equation and its applications, (Russian) Dokl. Akad. Nauk 435 (2010), no. 3, 310-313; English trans-lation in Dokl. Math. 82 (2010), no. 3, 909-911.
[80] M. B. Muratbekov and M. Otelbaev, On the smoothness of solutions of nonlinear Sturm-Liouville equation, (Russian) Abstracts Kazakhstan Interuniversity Conference on Mathematics and Mechanics, Karaganda, (1981), 34-35.
[81] K. T. Mynbaev and M. Otelbaev, Weighted functional spaces and the spectrum of differential operators, Nauka, Moscow, (1988).
[82] R. Oinarov, Separability of the Schr¨odinger operator in the space of
summable functions, (Russian) Dokl. Akad. Nauk SSSR 285 (1985), no. 5, 1062-1064; English translation in Soviet Math. Dokl. 32 (1985), no. 3, 837-839.
[83] S. Omran, Kh. A. Gepreel and E. T. A. Nofal, Separation of the general differential wave equation in Hilbert space, Int. J. Nonlinear Sci. 11 (2011), no. 3, 358-365.
[84] L. S. Ornstein and G. E. Uhlenbeck, On the theory of Brownian motion, Phys. Rev. 36 (1930), 823-841.
[85] K. N. Ospanov and R. D. Akhmetkaliyeva, Separation and the existence theorem for second order nonlinear differential equation, Electron. J. Qual. Theory Differ. Equ. (2012), no. 66, 12 pp.
[86] K. N. Ospanov and R. D. Akhmetkaliyeva, Some inequalities for second order differential operators with unbounded drift, Eurasian Math. J. 6 (2015), no.2, 63-74.
[87] M. Otelbaev, On the method of Titchmarsh for estimation of the resol-vent, (Russian) Dokl. Akad. Nauk SSSR 211 (1973), 787-790; English translation in Soviet Math. Dokl. 14 (1973), 1120-1124.
[88] M. Otelbaev, The summability with weight of the solution of a Sturm-Liouville equation, (Russian) Mat. Zametki 16 (1974), 969-980; English translation in Math. Notes 16 (1974), no. 6, 1172-1179.
[89] M. Otelbaev, Estimates of the eigenvalues of singular differential oper-ators, (Russian) Mat. Zametki 20 (1976), no. 6, 859-867; English trans-lation in Math. Notes 20 (1976), no. 5-6, 1038-1042.
[90] M. Otelbaev, The separation of elliptic operators, (Russian) Dokl. Akad. Nauk SSSR 234 (1977), no. 3, 540-543; English translation in Soviet Math. Dokl. 18 (1977), no. 3, 732-736.
[91] M. Otelbaev, The smoothness of the solution of differential equations, (Russian, Kazakh summary) Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. (1977), no. 5, 45-48.
[92] M. Otelbaev, A criterion for discreteness of the spectrum for a certain degenerate operator, and certain imbedding theorems, (Russian) Differ-encial’nye Uravnenija 13 (1977), no. 1, 111-120, 190.
[93] M. Otelbaev, Estimates of the spectrum of elliptic operators and em-bedding theorems associated with them, (Russian) Thesis for the degree of doctor of physical and mathematical sciences, Moscow, (1979). [94] M. Otelbaev, Imbedding theorems for spaces with a weight and their
application to the study of the spectrum of a Schr¨odinger operator,
(Russian) Trudy Mat. Inst. Steklov 150 (1979), 265-305.
[95] M. Otelbaev, Imbedding theorems for weighted spaces with a weight
and their application to the study of the spectrum of the Schr¨odinger
operator, Proc. Steklov Inst. Math. 4 (1981), 281-321.
[96] M. Otelbaev, Coercive estimates and separability theorems for elliptic
equations in Rn, (Russian) Trudy Mat. Inst. Steklov 161 (1983),
195-217.
[97] M. Otelbaev, Estimates of the spectrum of the Sturm-Liouville operator, (Russian), Gylym, Almaty (1990).
[98] G. Da Prato and A. Debussche, An integral inequality for the invariant measure of a stochastic reaction-diffusion equation, J. Evol. Equ. 17 (2017), no. 1, 197-214.
[99] G. Da Prato and B. Goldys, Elliptic operators on Rd with unbounded
xxxvi BIBLIOGRAPHY
[100] G. Da Prato and V. Vespri, Maximal Lpregularity for elliptic equations
with unbounded coefficients, Nonlinear Anal. 49 (2002), no. 6, Ser. A: Theory Methods, 747-755.
[101] P. J. Rabier, Elliptic problems on RN with unbounded coefficients in
classical Sobolev spaces, Math. Z. 249 (2005), no. 1, 1-30.
[102] D. Z. Raimbekov, Smoothness of the solution in L2 of a singular
equa-tion, (Russian) Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. (1974), no. 3, 78-83.
[103] M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier Analysis, self-adjointness, Academic Press, New York-London, (1975), 361 pp.
[104] M. Sapenov and L. A. Shuster, On the summability with weight of the solutions of binomial differential equations, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. (1987), no. 1, 38-42.
[105] E. I. Shemyakin, Propagation of nonstationary perturbations in a visco-elastic medium, Dokl. Akad. Nauk SSSR 104, (1955), no.1, 34-37. [106] B. Ya. Skachek, Distribution of eigenvalues of multidimensional
differ-ential operators, (Russian) Funkt. Anal. i Ego Prilozh., 9 (1975), no. 1, 83-84.
[107] M. Sobajima, Lp -theory for second-order elliptic operators with
un-bounded coefficients in an endpoint class, J. Evol. Equ. 14 (2014), no. 2, 461-475.
[108] A. N. Tikhonov and A. A. Samarskii, Equations of mathematical physics, Macmillan, New York (1963).
[109] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations, Clarendon Press 2 (1950).
[110] A. Zh. Togochuev, Summability of the solution of a differential equation of odd order with weight, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. (1985), no. 5, 55-58.
[111] S. S. Voit, Propagation of an initial condensation in viscous gas, (Rus-sian) Doklady Akad. Nauk SSSR 88, (1953), 221-224.
[112] M.C. Wang and G.E. Uhlenbeck, On the theory of the Brownian mo-tion, II, Rev. Modern Phys. 17 (1945), 323-342.
[113] E. M. E. Zayed, A. S. Mohamed and H. A. Atia, Separation for
Schr¨odinger-type operators with operator potentials in Banach spaces,
Appl. Anal. 84 (2005), no. 2, 211-220.
[114] E. M. E. Zayed, A. S. Mohamed and H. A. Atia, On the separation of el-liptic differential operators with operator potentials in weighted Hilbert spaces, Panamer. Math. J. 15 (2005), no. 2, 39-47.
[115] A. Zettl, Separation for differential operators and the Lp spaces, Proc.
Separation and the existence theorem for second order nonlinear differential equation1
K.N. Ospanov2and R.D. Akhmetkaliyeva
L.N. Gumilyov Eurasian National University, Kazakhstan kordan.ospanov@gmail.com, raya 84@mail.ru
Abstract. Sufficient conditions for the invertibility and separability in L2(−∞, +∞) of the degenerate second order differential operator with complex-valued coefficients are obtained, and its applications to the spectral and approximate problems are demonstrated. Using a separability theorem, which is obtained for the linear case, the solvability of nonlinear second order differential equation is proved on the real axis. Keywords: separability of the operator, complex-valued coefficients, completely continuous resolvent Mathematics subject classifications: 34B40
1. Introduction and main results
A concept of the separability was introduced in the fundamental paper [1]. The Sturm-Liouville’s operator
Jy = −y′′+ q(x)y, x ∈ (a, +∞),
is called separable [1] in L2(a, +∞), if y, −y′′+ qy ∈ L2(a, +∞) imply −y′′, qy ∈
L2(a, +∞). From this it follows that the separability of J is equivalent to the existence
of the estimate ky′′k
L2(a,+∞)+ kqykL2(a,+∞)≤ c
kJykL2(a,+∞)+ kykL2(a,+∞)
, y ∈ D(J), (1.1)
where D(J) is the domain of J. In [1] (see also [2, 3]) some criteria of the separability depended on a behavior q and its derivatives has been obtained for J. Moreover, an example of non-separable operator J with non-smooth potential q was shown in this papers. Without differentiability condition on function q the sufficient conditions for the separability of J has been obtained in [4, 5]. In [6,7] so-called Localization Principle of the proof for the separability of higher order binomial elliptic operators was developed in Hilbert space. In [8,9] it was shown that local integrability and semiboundedness from below of q are enough for separability of J in L1(−∞, +∞). Valuation method of
Green’s functions [1-3,8,9] (see also [10]), parametrix method [4,5], as well as method of local estimates for the resolvents of some regular operators [6, 7] have been used in these works.
Sufficient conditions of the separability for the Sturm-Liouville’s operator y′′+ Q(x)y
have been obtained in [11-15], where Q is an operator. A number of works were devoted to the separation problem for the general elliptic, hyperbolic and mixed-type operators. An application of the separability estimate (1.1) in the spectral theory of J has been shown in [15-18], and it allows us to prove an existence and a smoothness of solutions of nonlinear differential equations in unbounded domains [11, 17-20]. Brown [21] has shown that certain properties of positive solutions of disconjugate second
1Supported by L.N. Gumilyev Eurasian National University Research Fund. 2Corresponding author.
order differential expressions imply the separation. The connection of separation with concrete physical problems has been noted in [22].
We denote L2 := L2(R), R = (−∞, +∞), the space of square integrable functions.
Let l is a closure in L2 of the expression l0y = −y′′+ r(x)y′+ s(x)¯y′defined in the set
C∞
0 (R) of all infinitely differentiable and compactly sapported functions. Here r and s
are complex - valued functions, and ¯y is the complex conjugate to y.
In this report we investigate some problems for the operator l. Although the operator l, similarly to the Sturm-Liouville operator J, is a singular differential operator of second order, their properties are different. The theory of the Sturm-Liouville operator J, in contrast to the operator l, developing a long time, while the idea of research is often based on the positivity of the potential q(x) (see, eg, [1-20]). Because of the coefficients r and s, are the methods developed for the Sturm-Liouville problems are often not applicable to the study of the operator l. The spectral properties for self-adjoint singular differential operators of second order, without the free term, have been to a certain extent investigated; a review of literature can be found in [23, 24]. Note that the differential operator l is used, in particular, in the oscillatory processes in the medium with resistance depended on velocity [25, pp. 111-116].
The operator l is said to be separable in L2if the following estimate holds:
ky′′k 2+ kry ′k 2+ ks¯y ′k 2≤ c (klyk2+ kyk2) , y ∈ D(l),
where k·k2is the L2- norm. In the present communication the sufficient conditions for
the invertibility and separability of the differential operator l are obtained. Moreover, spectral and approximate results for the inverse operator l−1are achieved. Using a
sep-aration theorem, which is obtained for the linear case, the solvability of the degenerate nonlinear second order differential equation −y′′+ r(x, y)y′= F (x ∈ R) is proved.
Let’s consider the degenerate differential equation
ly = −y′′+ r(x)y′+ s(x)¯y′= f. (1.2)
The function y ∈ L2is called a solution of (1.2) if there exists a sequence {yn}+∞n=1such
that kyn− yk2 → 0, klyn− fk2→ 0 as n → +∞. If the operator l is separable, then
the solution y of (1.2) belongs to the weighted Sobolev space W2
2(R, |r| + |s|) with the
norm ky′′k
2+ k(|r| + |s|)y′k2. So, the study of the qualitative behavior of solutions of
(1.2) and spectral and approximative properties of l can be reduced to the investigation of embedding W2 2(R, |r| + |s|) ֒→ L2. We denote αg,h(t) = kgkL2(0,t)k1/hkL2(t,+∞)(t > 0), βg,h(τ ) = kgkL2(τ,0)k1/hkL2(−∞,τ )(τ < 0), γg,h= max sup t>0αg,h(t), supτ <0βg,h(τ ) ,
where g and h are given functions. By Cloc(1)(R) we denote the set of functions f such
that ψf ∈ C(1)(R) for all ψ ∈ C∞ 0 (R).
Theorem 1. Let functions r and s satisfy the conditions
r, s ∈ Cloc(1)(R), Re r − |s| ≥ δ > 0, γ1,Re r< ∞. (1.3)
