Open access peer-reviewed chapter

Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces

Written By

Bilal Bilalov, Sabina Sadigova and Zaur Kasumov

Reviewed: 13 April 2022 Published: 11 June 2022

DOI: 10.5772/intechopen.104918

From the Edited Volume

Nonlinear Systems - Recent Developments and Advances

Edited by Bo Yang and Dušan Stipanović

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Abstract

In this chapter an m-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. It is established an interior and up-to boundary Schauder-type estimates with respect to these Sobolev spaces for m-th order elliptic operators, the trace of functions and trace operator are determined, the boundedness of trace operator and the extension theorem are proved, the properties of the Riesz potential are studied regarding these Sobolev spaces, etc. It is considered a second-order elliptic equation, and we study the Fredholmness of the Dirichlet problem in the Sobolev space generated by a separable subspace of the grand Lebesgue space. It is also considered one spectral problem for a discontinuous second-order differential operator and proved the theorem on the basicity of eigenfunctions of this operator in subspace of Morrey space, in which the infinitely differentiable functions with compact support are dense.

Keywords

  • non-standard function spaces
  • grand-Sobolev spaces
  • space of traces
  • Schauder type estimates
  • Riesz potentials
  • elliptic equations
  • Fredholmness
  • spectral problem
  • basicity
  • Morrey space

1. Introduction

Differential (also elliptic) equations play an especial role in the study of various processes and phenomena in natural science. Solvability problems of elliptic equations have a very rich history and remarkable monographs by various famous mathematicians are devoted to them. The theory of elliptic equations was developed in an comprehensive way in Hölder classes (solution in the classical sense) and in Hilbertian Sobolev spaces W2kΩ. In the above mentioned case, depending on the nature of the problem, there are various methods of solution (for instance, the method of potentials, the periodic case, the method of the theory of functions, spectral method, etc.), which cannot be said for the non-Hilbert case WpkΩ,p2, in which each method faces certain difficulties. All considered spaces are separable Banach spaces and infinitely differentiable and finite functions are dense in them. In the study of solvability of differential equations these facts are significant. Note that one of the methods to solve differential equations is a spectral method. To justify the solution by this method, one should study the basis properties of the root vectors of the considered spectral problem in the appropriate Banach function space.

In connection with applications in problems of mechanics, mathematical physics and pure mathematics, the so-called non-standard spaces of functions have greatly increased and the list of such spaces includes Lebesgue spaces with a variable summability index, Morrey spaces, grand Lebesgue spaces, Orlicz spaces, etc. For more details one can see the monographs [1, 2, 3, 4, 5, 6]. Compared with other areas of mathematics, the apparatus of harmonic analysis has been fairly well studied in relation to these spaces. The problems of analysis and approximation theory have been relatively well studied in Lebesgue spaces with variable summability index and Morrey spaces (see [7, 8, 9, 10, 11, 12, 13, 14]). The above mentioned problems have begun to be studied in Grand Lebesgue spaces, and valuable results have been obtained in this direction (see [15, 16]). The solvability problems of partial differential equations have also begun to be studied in the Sobolev spaces generated by these spaces (see [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]). Morrey-Sobolev and grand-Sobolev spaces are not separable and therefore infinitely differentiable and finite functions are not dense in them, in this reason the study the problems of solvability of differential equations in these spaces are of special scientific interest. Therefore, it is necessary to extract reasonable subspaces dictated by differential equations and develop an instruments for studying the solvability of differential equations in these subspaces.

An m-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. It is established an interior and up-to boundary Schauder-type estimates with respect to these Sobolev spaces for m-th order elliptic operators, the trace of functions and trace operator are determined, the boundedness of trace operator and the extension theorem are proved, the properties of the Riesz potential are studied regarding these Sobolev spaces, etc. It is considered a second-order elliptic equation and we study the fredholmness of the Dirichlet problem in the Sobolev space generated by a separable subspace of the grand Lebesgue space. It is considered one spectral problem for a discontinuous second order differential operator and proved the theorem on the basicity of eigenfunctions of this operator in subspace of Morrey space, in which the infinitely differentiable functions with compact support are dense.

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2. Needful information

2.1 Standard notation

Z+ will be the set of non-negative integers. 1,n̅=1,2,,n. C is the set of complex numbers. Brx0=xRn:xx0<r will denote the open ball in Rn centered at x0, where x=x12++xn2,x=x1xn. ·· is a scalar product in Rn. mesM will stand for the Lebesgue measure of the set M; ∂Ω will be the boundary of the domain Ω; Ω¯=Ω∂Ω; diamΩ will stand for the diameter of the set Ω; f/M denotes the restriction of f to M. C0Ω will denote the space of infinitely differentiable and finite functions in Ω and CmΩwill stand for the space of m-th order continuously differentiable functions in the domain Ω. C0mΩwill stand for the space of m-th order continuously differentiable and finite functions in the domain Ω. DLwill stand for the domain of the operator L; RTwill stand for the range of the operator T; KerT is the kernel of the operator T; T is the adjoint of T; XY is a Banach space of bounded operators acting from X to Y; X=XX. Throughout this paper, q will denote the conjugate of a number, i.e. 1q+1q=1.

2.2 Elliptic operator of m-th order and some necessary facts

Let ΩRn be some bounded domain with the rectifiable boundary ∂Ω. We will use the notations of [19]. α=α1αn will be the multiindex with the coordinates αkZ+,k=1,n̅; i=xi will denote the differentiation operator, α=1α12α2nαn. For every ξ=ξ1ξn we assume ξα=ξ1α1ξ2α2ξnαn. Let L be an elliptic differential operator of m-th order

L=pmapxp,E1

where p=p1pn, pkZ+,k=1,n̅,ap·LΩ are real functions. Consider the elliptic operator L0:

L0=p=map0p,E2

with the constant coefficients ap0 and denote by J· a fundamental solution of Eq. (2) [28].

Let L be an elliptic operator and consider a “tangential operator”

Lx0=p=mapx0p,E3

at every point x0Ω. Denote by Jx0· the fundamental solution of the equation Lx0φ=0. The function Jx0· is called a parametrics for the equation =0 with a singularity at the point x0. Let

Sx0φ=ψx=Jx0xyφydy,E4

and

Tx0=Sx0Lx0L.E5

Denote the operators Sx0,Lx0 and Tx0, corresponding to the point x0=0, by S0,L0 and T0, respectively.

Let us give the definition of smooth boundary.

Definition 1.1 We will say that the boundary Ω of a domain ΩRn belongs to class Ck if each sufficiently small piece of it can be mapped onto a segment of the hyperplane xn=0 using a coordinate transformation yx=y1xynx with a positive Jacobian so that yiCk,i=1,n̅.

2.3 Grand-Sobolev spaces Wq)mΩand WNq)mΩ

Define the grand-Lebesgue space Lq)Ω, 1<q<+ (throughout this paper we will assume that this condition holds on q). Grand-Lebesgue space Lq)Ωis a Banach space of (Lebesgue) measurable functions f on Ω with the norm

fq)=sup0<ε<q1εΩfqεdx1qε.E6

The following continuous embeddings hold

LqΩLq)ΩLqεΩ,E7

where ε0q1 is an arbitrary number. The space Lq)Ω is not separable.

Below in this section we will assume that every function defined on Ω is extended by zero to Rn\Ω¯. Let Tδ be a shift operator, i.e. Tδfx=fδ+x,xΩ and δRn. Let

Nq)Ω=fLq)Ω:Tδffq)0δ0.E8

The space Nq)Ω is a Banach space with the norm ·q), (i.e. is the subspace of Lq)Ω.)

The following lemma is true (see [17]).

Lemma 1.2 C0Ω̅= Nq)Ω (the closure is taken in Lq)Ω).

Let us include the following lemma without proof.

Lemma 1.3 The embeddings

LqΩNq)ΩLq)ΩL1Ω,E9

hold and every inclusion is strict.

Denote by Wq)mΩ the grand-Sobolev space generated by the norm

fWq)m=k=0mfkq).E10

Let

WNq)mΩ=fWq)mΩ:TδffWq)m0δ0.E11

Consider the following singular kernel

kx=ωxxn,E12

where ωx is a positive homogeneous function of degree zero, which is infinitely differentiable and satisfies

x=1ωx=0,E13

being a surface element on the unit sphere. By K we will denote the corresponding singular integral

Kfx=kfx=Ωfykxydy.E14

The following theorem is valid for the operator K (see [4]).

Theorem 1.4 [4] The inclusion KLq)Ω,1<q<+ is valid, i.e. c>0:

Kfq)cfq),fLq)ΩE15

The validity of the following lemma is given in [17].

Lemma 1.5 [17] Nq)Ω,1<q<+, is an invariant subspace of the singular operator K in Lq)Ω.

Considering the expression for the norm Nq), it is not hard to prove the following.

Proposition 1.6 Let ΩRnbe a bounded domain and L be an elliptic operator with coefficients apLΩ,pm. Then it is valid LWNq)mΩNq)Ω, i.e. the following inequality

LuNq)ΩCuWNq)mΩ,uWNq)mΩ,E16

holds, where C>0 is a constant independent of u.

In the sequel, when Ω=Br the spaces Lq)Ω,Nq)Ω,Wq)Ω and WNq)Ω will be redenoted by Lq)r,Nq)r,Wq)mr and WNq)mr, respectively. Along with WNq)mΩ, consider the following space of functions Nq)mΩ equipped with the norm

fNq)mΩ=pmdΩpnqpfLq)Ω,E17

where dΩ=diamΩ, and we will assume Nq)0Ω=Nq)Ω. The closure of C0Ω in Nq)mΩ (NqmΩ) we will denote by Nq)mΩ (NqmΩ).

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3. Main lemma

3.1 Solvability in the small

Introduce the following

Definition 1.7 We will say that the operator L has the property Px0) if its coefficients satisfy the conditions: i) apLBrx0, pm, for some r>0; ii) r>0: for p=m the coefficient ap· coincides a.e. in Brx0 with some function bounded and continuous at the point x0.

It is absolutely clear that if apCΩ, pm, then L has the property Px0) for x0Ω.

In establishing the interior Schauder-type estimate for grand-Sobolev spaces Nq)mΩ the following Main Lemma, proved in [18] plays a key role.

Main Lemma. Let the m-th order elliptic operator L have the property Px0) at the point x0. Let φNq)mBrx0 and φ vanish in a neighborhood of xx0=r. Then for q>1 it holds

Tx0φNq)mBrx0σrφNq)mBrx0,E18

where the function σr0,r0, depends only on the ellipticity constant Lx0, on the coefficients of L and their moduli of continuity.

Let us consider the m-th order elliptic operator L with the coefficients apx defined by (1), and the corresponding operator Tx0 defined by (5). Using Main Lemma, it is proved the following local existence theorem.

Theorem 1.8 Let L be an m-th order elliptic operator which has the property Px0) at some point x0Ω and fGq)Ω,1<q<+. Then, for sufficiently small r, there exists a solution of the equation Lu=f belonging to the class Nq)Brx0.

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4. Interior Schauder type estimates

Let ω· be an infinitely differentiable function on 01 such that for 0t<13, ωt1 and for 23<t1, ωt0. For 0<R1<R2 we put

ξx=1,xR1,ωxR1R2R1,R1<xR2.E19

Regarding this function it holds the following.

Lemma 1.9 There is a constant C>0 depending only on R2 and ω·, such that for R1:0<R1<R2, there is

ξCmR2C1R1R2m.E20

Accept the following property with respect to the domain Ω.

Property α).We say that the domain Ω admits the continuation of functions of the space Nq)kΩ if there exists a domain ΩΩ¯ and a linear mapping θ of the space Nq)kΩ into Nq)kΩ such that

θu=uinΩ,
θuNq)kΩconstuNq)kΩ,E21

holds.

So, the following lemma is true.

Lemma 1.10 Let the domain Ω have the Property α) with respect to space Nq)kΩ. Then C>0 depending only on n,q and on a constant from (21), which holds

φNq)kΩεφNq)k+1Ω+CεkφLq)Ω,E22

for k=1,m1̅ and ε>0.

The main result of this section is the following Schauder type estimate.

Theorem 1.11 Let the coefficients of m-th order elliptic operator L satisfy the following conditions: i) ap·CΩ¯, p:p=m; ii) ap·LΩ, p:p<m; where ΩRn bounded domain with boundary Ω. Let Ω0Ω be an arbitrary compact. Then for uWGq)mΩ, 1<q<+, the following a priori estimate holds

uNq)mΩ0CLuLq)Ω+uLq)Ω,E23

where the constant C depends only on the ellipticity constant m,Ω,Ω0 of L, on the coefficients of the operator L.

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5. Extension of functions from NqmΩCompactness

Consider the question of the possibility of extension of a function f from class NqmΩ to a wider class NqmΩ with ΩΩ¯. Following the classical case (see, monograph [29]), first consider the case when Ω is a cube with an edge 2a>0:Ka=yi<ai=1,n̅, and Ω is a parallelepiped Ka+=Kayn>0 .

The following lemma is true.

Lemma 1.12 For fWNq)kKa+ there exists an extension FWNq)kKa and, in addition, inequality

FNqmKaCfNqmKa+,E24

holds.

It is completely analogous to the monograph [29, p. 129], it is proved the following

Lemma 1.13 Let fWNq)kΩ and for ξΩ there exists a function Fξx, defined in a ball Brξ of some radius r=rξ>0, such that Fξx=fx,xΩBrξ and FξWNq)kBrξ. Besides

FξWNq)kBrξCfWNq)kΩ,E25

is true, where C>0 is a constant independent of f. Then, for any ρ>0, the function f has an extension F to the domain Ωρ=xΩBρx with the properties FWNq)kΩρ,Fx=0,xΩρ\Ωρ/2: and the inequality

FWNq)kΩρCfWNq)kΩ,E26

holds, where the constant C>0, is dependent only on domain Ω and ρ.

Using Main Lemma and Lemmas 1.12, 1.13, similarly to [29, p. 130] it is proved the following extension.

Theorem 1.14 Let Ω, Ω be bounded domains in Rn, Ω¯Ω and ΩCm. Then for fNqmΩ there exists a finite extension FNqmΩ in Ω and the following estimate

FNq)mΩCfNq)mΩ,E27

is valid, where C>0 is a constant independent of f.

Consider the compactness question of the family in NqΩ. The following theorem is true.

Theorem 1.15 Let ΩRn be a bounded domain with a boundary ΩC1. Then a set, bounded in Nq1Ω, is compact in NqΩ.

Analogously to Theorem 1.15, the following theorem is also proved.

Theorem 1.16 Let ΩRn be a bounded domain with a boundary ΩCk. If the set of functions is bounded in NqkΩ, then the set of their traces on n1-dimensional surface ΓΩ¯ from the class Ck is compact in WqrΩ, r=0,k1̅.

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6. Trace of functions from the grand-Sobolev space Nq)mΩ

In this section, we will define a concept of the trace for functions from the grand-Sobolev space Nq)mΩ on an n1-dimensional differentiable surface.

Based on the embedding Nq)mΩNq)1Ω,m2, it is sufficient to define this concept regarding the functions from Nq)1Ω. So, assume SΩ¯: SC1 is some n1-dimensional surface. Let x0S be an arbitrary point. Then it is obvious that there exists a sufficiently small neighborhood of this point Sx0S, such that uniquely projected onto some domain D of the plane xn=0 in Rn and it has the equation

xn=φxC1D¯,x=x1xn1D.E28

Ω is bounded domain and we will consider that it is placed inside a cube 0<xi<ai=1,n̅, with an edge a>0. Let fCΩ be some finite function in Ω. For xφxSx0 we have

fx/Sx0=fxφx=0φxfxξnξndξn.E29

Let ε0q1be an arbitrary number, qε=qε and 1qε+1qε=1. Applying Hölder’s inequality from (29), we obtain

f/Sx0qεφxqεqε0φxfxξnξnqεdξnaqεqε0φxfxξnξnqεdξn.E30

Let C=maxa1. Consequently

εf/Sx0qεCqεε0φxfxξnξnqεdξn,E31

where C is a constant independent of f and ε. Multiplying by 1+φx12++φxn12 and integrating over D we obtain

ε1qεfLqεSx0qεCε1qεfxnLqεΩqε.E32

Since the surface S can be covered by a finite number of surfaces of type Sx0, then summing the corresponding inequalities (from (32)) we establish

ε1qεfLqεSqεCε1qεfxnLqεΩqε,E33

where C>0 is a constant independent of f and ε. This immediately implies

ε1qεfLqεSCp=1dΩε1qεpfLqεΩCε1qεpfNqε1Ω.E34

Taking first sup0<ε<q1 on the right and then the same sup on the left, from this estimate for fC0Ω, we have

fLqSCfNq1Ω.E35

If ∂ΩC1, then Theorem 1.14 implies that the inequality (35) holds for fC1Ω¯.

Let fNq1Ω be an arbitrary function. Then fnCΩ¯:

fnfNq1Ω0,n.E36

It follows directly from (35) that the sequence fn is fundamental in LqS:

fnfmLqS0,n,m.E37

From the completeness of LqS it follows that fSLqS:

fnfSLqS0,n.E38

Similarly the classical case, it is proved that fS does not depend on the choice of the sequence fn.

fS is called the trace of the function fNq1Ω on S and we will denote it by the operator ΓS: Γf=f/S.

Based on the concept of ΓS, we define the following linear space

Nq1S=Nq1Ω/S=fLqS:uNq1Ωf=ΓSu=u/S.E39

For the case of S=∂Ω, the operator ΓS will be simply denoted by Γ:Γ∂Ω=Γ.

The following lemma is true.

Lemma 1.17 Let ΩRn be an bounded domain and ΩC1. Then the linear spaces Fq)1 and Nq1Ω are isomorphic, where

Nq1∂ΩNq1Ω/∂Ω=fLq∂Ω:uNq1Ωf=Γu=u/∂Ω.E40

Based on this lemma, we define the norm in Nq1∂Ω

fNq1∂Ω=Γ1fFq1,fNq1∂Ω.E41

Since Fq1 is a Banach space with respect to the norm ·Fq1, then this lemma immediately implies that Nq1∂Ω is also Banach with respect to the norm (41).

The space Nqm∂Ω is defined similarly and the corresponding lemma is true for the spaces Fq)m, where

Nqm∂ΩNqmΩ/∂Ω=fLq)∂Ω:uNqmΩf=Γu=u/∂Ω.E42

The following theorem is true (regarding the proof see [30]).

Theorem 1.18 Let ΩRn be a bounded domain with a boundary ΩCm. If the set of functions is bounded in Nq)mΩ,m1, then the set of their traces on the n1-dimensional surface SΩ¯ from the class Cm is compact in Lq)S.

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7. Schauder-type estimate up to the boundary

Using the results obtained in the previous sections, it is established a Schauder-type estimate up to the boundary for a second-order elliptic operator with nonsmooth coefficients. The following theorem is true.

Theorem 1.19 Let ΩRn be a bounded domain with a boundary ΩC2 and L be a second-order elliptic operator (i.e. m=2) with coefficients apCΩ¯,p:p=m and apLΩ,p:p<m. Then for uNq2Ω the following estimate

uNq2ΩCLuNqΩ+uNq2∂Ω+uNqΩ,E43

holds true, where C>0 is a constant independent of u, but depends on the norms of the coefficients of L in LΩ.

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8. Solvability of the Dirichlet problem for a second-order elliptic operator

Let us apply the estimates established in the previous sections to the solvability question (in the strong sense) of the Dirichlet problem for a second-order elliptic type equation in classes Nq2Ω. So, let ΩRn be a domain with a boundary ΩC2 . Assume that fNqΩ is a given function and aij CΩ¯; ai;aLΩ i;j=1,n̅. Consider the equation

Lu=i;j=1naijx2uxixj+i=1naixuxi+axu=fx,xΩ.E44

In the sequel we will assume that the following uniformly ellipticity condition holds a.e. in Ω

νξ2i,j=1naijxξiξjν1ξ2,ξRn,E45

where ν01 is some constant.

Under the solution of the Eq. (44) we mean a function uNq2Ω for which equality (44) holds a.e. xΩ. Let φNq2∂Ω be a given function. Let us define the boundary condition

Γu=u/∂Ω=φE46

where Γ:Nq2ΩNq2∂Ω is a trace operator.

We will say that the domain Ω has a property δq is class Nq)2Ω, if the Dirichlet problem (46) is correctly solvable for the Poisson equation, i.e. the problem

Δu=f,inΩ,Γu=0,on∂Ω,E47

has a unique solution for fNq)Ω in class Nq)2Ω.

In order to solve the problem (44), (46) we apply the parameter continuation method (see e.g. [28, p. 247]).

Furthermore, assume that the operator L satisfies the following inequality

uLqΩCLuLqΩ,uNq)2Ω,E48

where the constant C depends only on the ellipticity constants of the operator L, on the sup norms of the coefficients L, on domain Ω and is independent of the function uNq)2Ω.

We will say that the operator L has property (A) if an inequality (48) holds for an operator L.

The question of whether inequality (48) holds (i.e. property (A)) we will consider later.

Thus, the following main theorem is true.

Theorem 1.20 Let ΩRn be a bounded domain with a boundary ΩC2 and L is a second-order elliptic differential operator defined by expression (44) with coefficients aijCΩ¯; ai; aLΩ, i;j=1,n̅. Assume that the domain Ω has property Δq) in class Nq)2Ω and the operator L has property (A). Then the equation Lu=f is uniquely solvable for fNqΩ in class Nq)2Ω, i.e. L:Nq)2ΩNqΩ is an isomorphism and it is obvious that the estimate

uNq)2ΩCfNq)Ω,fNq)Ω,E49

holds true, where C>0 is a constant independent of f.

Now consider a homogeneous equation Lu=0 in Ω with a nonhomogeneous boundary condition Γu=u/∂Ω=φ, where φNq)2Ωis given function. From the results of Section 6 it follows that ΦNq)2Ω:ΓΦ=φ. Suppose υ=uΦ and let f=LΦ. It is clear that Γυ=υ/∂Ω=0 and =f in Ω. If aij;ai; aLΩ, i;j=1,n̅, then, as follows from Proposition 1.6 that fNq)Ω. Therefore, we can apply the Theorem 1.20 to the problem

=f,a.e.inΩ,υ/∂Ω=0.E50

If all the conditions of Theorem 1.20 are satisfied, then this problem is uniquely solvable in class Nq)2Ω and for the solution it is valid the following estimate

υNq)2ΩCfNq)Ω,E51

where C>0 is a constant independent of f. It is quite obvious that then the problem

Lu=0,a.e.inΩ,u/Γ=φ,E52

is also uniquely solvable in Nq)2Ω.

Taking into account expression (41) for the norm in Nq)2∂Ω, we obtain

uNq)2ΩCφNq)2∂Ω.E53

Consider a nonhomogeneous equation with a nonhomogeneous boundary condition

Lu=fa.e.inΩ,u/∂Ω=φ,E54

where fNq)Ω and φNq)2∂Ωare given functions. Representing the function u in the form u=v+w, where

Lv=f,v/∂Ω=0,Lw=0,w/∂Ω=φ,E55

from Theorem 1.20 and taking into account estimate (53), we arrive at the following conclusion.

Theorem 1.21 Let the domain Ω and the operator L satisfy all the conditions of Theorem 1.20. Then for fNq)Ω and φNq)2Ω the problem (54) is uniquely solvable in the space Nq)2Ωand regarding the solution the estimate

uNq)2ΩCfNq)Ω+φLq)∂Ω,E56

is fulfilled, where C>0 is a constant independent of f and φ.

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9. Some properties of a Riesz potential

For obtaining main results we need some properties of a Riesz potential and embedding theorems regarding the spaces Nq)m. In this section, we will give some properties of an integral operator with a weak singularity. These properties are used to study the properties of functions from class WqkΩ. Let us remember the Sobolev integral identity

ux=α=0k1xαΩbαyuydy+α=kΩAαxyrnkαuydy,uCkΩ¯,E57

where bαCΩ¯, ALΩ×Ω (generally speaking, for xy: Axy is infinitely differentiable). In establishing many properties of a function from Sobolev classes the representation (57) plays a key role. In accordance with (57) consider the integral operator (Riesz potential).

x=Vx=ΩAxyrλρydy,E58

where r=yx; xΩRn is a bounded domain, 0λ<n; ALΩ×Ω. The following theorem is true.

Theorem 1.22 Let ΩRn be a bounded domain, ρLqΩ, ACΩ¯×Ω¯ and λq<n. Then operator (58) acts compactly from Lq)Ω to CΩ¯.

It is true the following classical analogue

Theorem 1.23 Let λqn and an integer s satisfy nnλ<sn. Then the integral (58) defines a function that, on any intersection Ωs of the set Ω by a plane of dimension s, is defined almost everywhere in the sense of the Lebesgue measure in Rs. The operator K defined by formula (58) is bounded as an operator from LqΩ to LrΩs (also from LqΩ to LrΩs), for r: 1<r<r0=sqnnλq.

It is valid the following

Theorem 1.24 If λqn, then the operator K, defined by expression (58), acts compactly from LqΩ to LrΩ (also from Lq)Ω to LrΩ), for r:1<r<r0=nqnnλq.

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10. Embedding theorems

To obtain Sobolev-type embedding theorems in spaces WqkΩ we will use the results obtained in the previous section. Throughout this section, we assume that the domain ΩRn is bounded and stellar relative to some sphere. Remember that a domain is called stellar relative to some point if any ray outgoing from this point has one and only one common point with the boundary of this domain. A domain is stellar with respect to some set if it is stellar at every point of this set. The following theorem is true.

Theorem 1.25 If qk>n, then WqkΩ compactly embedded in CΩ¯.

The following theorem is true.

Theorem 1.26 Let qkn and ΩsΩbe a piecewise smooth manifold of s dimensions, where nkq<sn. Then WqkΩ continuously embedded in LrΩs (also in Lr)Ωs), where 1<r<r0=sqnkq.

The following theorems are proved in a completely similar way.

Theorem 1.27 If qkn, then WqkΩ compactly embedded in LrΩ (also in Lr)Ω), where 1r<nqnkq .

The following theorem is also true.

Theorem 1.28 Let uWqkΩ. Then it has all possible generalized derivatives of any order l<k in Ω. At the same time WqkΩ compactly embedded in ClΩ¯, if klq>n and in WrlΩ(also in Wr)lΩ), if klqn and 1r<nqnkq.

Let us give some equivalent norms in the grand-Sobolev spaces WqkΩ (then in NqkΩ). Let a function f· continuous in Rr have the following properties

α)ft0ft=0t=0;
β)fλt=λft,λRtRr;E59
γ)ft+τft+fτ,t;τRr.

In a completely analogous way to the classical case, the following theorem is proved.

Theorem 1.29 Let r denote the number of distinct monomials of degree k1 and let l1,,lr be linear functionals bounded on WqkΩ that do not simultaneously vanish on any polynomial of degree k1, except for the identically zero. Let f· be a continuous function in Rr, having the properties of a norm α)γ). Then the norm

uq,k=fl1ul2ulru+α=kαuLqΩ,E60

is equivalent to the norm ·WqkΩ.

The following lemma is true.

Lemma 1.30 Let ΩRn be a bounded domain with sufficiently smooth boundary Ω. Then the norm ·q,k, defined by expression

uq,k=α=0k1∂Ωαudσ+α=kαuLqΩ,E61

is equivalent to ·WqkΩ in WqkΩ. In this case for uWq)kΩ the following inequality

uLqΩCα=kαuLqΩ,E62

holds, C is a constant independent of u.

11. About the property (a). Fredholmness

Let us get back to the question of whether property (A) is satisfied. Let ΩRn be a bounded domain. Let SΩ¯ be some n1-dimensional surface. Define the following class of functions. Let SΩ¯ belong to class Ck and ε>0 be some number. Put ΩεS=xΩ:ρxS>ε.

We say that the function f belongs to class C0kS (i.e. f vanishes in some neighborhood S), if fCkΩ¯ and ε>0:fx=0, xΩ\ΩεS. Denote by Nq)kΩS the closure C0kS in NqkΩ.

Thus, it is clear that Nq)kΩ∂Ω=Nq)kΩ. Denote by FqmΩS the factor space NqmΩ/Nq)mΩS. Thus, FqmΩ∂Ω=Fqm. Each function fNq1Ω (also fNqkΩ) has a (unique) trace f/S on S, and f/SLqS. Consider the following class of functions

NqkS=NqkΩ/S=fLqS:uNqkΩf=u/S.E63

The following theorem is true.

Lemma 1.31 Let SΩ¯SCk be n1-dimensional surface. Then the linear spaces FqkΩS and NqkS, k1, are isomorphic.

It is not hard to see that if fNq)kΩS, then f/S=0, k1. Applying this lemma, completely similar to Theorem 1.19, we can prove the following.

Theorem 1.32 Let ΩRn be a bounded domain with a boundary ΩC2. Let L be a second-order elliptic operator with coefficients aijCΩ¯, ai,a LΩ, i, j=1,n¯ defined by expression (44). Let SΩ¯SC2 be some n1-dimensional surface and ΩΩS (i.e. Ω¯ΩS) be an arbitrary domain. Then the following estimate holds true for uNq)2ΩS:

uNq2ΩCLuNqΩ+uNqΩ,E64

where the constant C depends only on the ellipticity constants of operator L, on the norms of the coefficients of L in LΩ, on S and Ω (is independent of u).

It is not hard to see that Theorem 1.19 is a particular case of this theorem, for this it is sufficient to take S=∂Ω. By Theorem 1.32 completely analogous to Theorem 9.14 of the monograph [31, p. 240] the following theorem is proved.

Theorem 1.33 Let ΩRn be a bounded domain with a boundary ΩC2 and L be an elliptic operator (44) with coefficients aijCΩ¯, ai,a LΩ, i,j=1,n¯ . Then the following estimate holds for uNq)2Ω:

uNqΩCLuσuNqΩ,E65

for σσ0, where C; σ0>0 are some constants that independent of u.

The following theorem is the result of Theorems 1.20 and 1.33.

Theorem 1.34 Let L be an elliptic operator (44) with coefficients aijCΩ¯, ai, aLΩ, i,j=1,n̅. Let ΩRn be a bounded domain with a boundary ΩC2, which has a property Δq). Then σ0>0: the equation Luσu=f is uniquely solvable for fNqΩ in class Nq)2Ω, σσ0.

The Fredholm alternatives hold for the equation Lu=f, i.e. the following main theorem is true.

Theorem 1.35 Let L be an elliptic operator (44) with coefficients aijCΩ¯, ai, aLΩ, i,j=1,n̅ and ΩRn be a bounded domain with a boundary ΩC2, which has a property Δq). Then: i) if KerL=0 in Nq)2Ω, then the boundary value problem Lu=f, u/Γ=φ, has a unique solution for φFq2Ω and fNqΩ; ii) KerL is a finite-dimensional subspace in Nq)2Ω.

Regarding the proof of all these facts one can see the works [4, 6].

12. On one spectral problem in Morrey-Smirnov space

In this section we consider one spectral problem in Morrey-Smirnov space. Such spectral problems arise in the problem of vibrations of a loaded string with fixed ends is solved by applying the Fourier method (see [32, 33, 34]). Morrey space is also non separable space and we define its subspace in which the infinitely differentiable functions are dense. We prove that eigenfunctions of the considered spectral problem form a basis in this subspace after eliminating an arbitrary term from them.

We need some facts from the theory of Morrey-type spaces. Let Γ be some rectifiable Jordan curve on the complex plane C. By MΓ we denote the linear Lebesgue measure of the set MΓ. By the Morrey- Lebesgue space Lp,αΓ,0α1,p1, we mean a normed space of all functions f· measurable on Γ equipped with a finite norm fLp,αΓ:

fLp,αΓ=supBBΓΓα1BΓfξp1p<+.E66

Lp,αΓ is a Banach space and Lp,1Γ=LpΓ,Lp,0Γ=LΓ. The embedding Lp,α1ΓLp,α2Γ is valid for 0α1α21 . Thus Lp,αΓLpΓ,α01,p1. The case of Γab will be denoted by Lp,αab.

Denote by Lp,αab the linear subspace of Lp,αab consisting of functions whose shifts are continuous in Lp,αab, i.e. f·+δf·Lp,αab0 as δ0. The closure of Lp,αab in Lp,αab will be denoted by Mp,αab. In [35] the following theorem is proved.

Theorem 1.36 The exponential system eintnZ is the basis in Mp,αππ,1<p<+,0<α1.

Using this theorem, it is easy to obtain the following

Theorem 1.37 Each of the trigonometric systems sinnxn=1 and cosnxn=0 forms the basis for Mp,α0π,1<p<+,0<α1.

Consider a sample eigenvalue problem for the discontinuous second-order differential operator

yx=λyx,x013131,E67

with the boundary conditions

y0=y1=0,y130=y13+0,y130y13+0=λmy13,E68

where λ is the spectral parameter, m is a non-zero complex number.

Let us give some results from [36], which we will need throughout the paper.

Lemma 1.38 [36] The spectral problem (67), (68) has two series of eigenfunctions which are given by the following expressions

y1,nx=sin3πnx,x01,n=1,2,,E69
y2,nx=sinρ2,nx13+sinρ2,nx+13,x013,sinρ2,n1x,x131,n=0,1,2,.E70

Let us construct the operator L, which linearizes the problem (67), (68) in the direct sum Lp01C. Denote by Wp2013Wp2131 the space of functions whose restrictions to intervals 013 and 131 belong to Sobolev spaces Wp2013 and Wp2131, respectively, where 1<p<. Let us define the operator L in the following way. As its domain DL we take the manifold

DL=ŷ=yxmy13:yxWp2013Wp2131,y0=y1=0,y130=y13+0,E71

and for ŷDL the operator L is defined by the relation

Lŷ=yy130y13+0.E72

The following lemma holds true.

Lemma 1.39 The operator L defined by expressions (71), (72) is a densely defined closed operator with a completely continuous resolvent. The eigenvalues of the operator L and the problem (67), (68) coincide. If yx is the eigenfunction (associated function) of problem (67), (68), then ŷ=yxmy13 is the eigenvector (associated vector) of the operator L.

In order to obtain the main results, we need some concepts and facts from the theory of bases in a Banach space.

Recall the following definition.

Definition 1.40 The basis unnN of Banach space X is called a p-basis, if for any xX one has the inequality

n=1xϑnp1pMx,E73

where ϑnnN is the biorthogonal system for unnN.

Definition 1.41 The sequences unnN and φnnN of Banach space X are called p- close, if

n=1unφnp<.E74

We will also use the following results from [37, 38] (see also [39, 40]).

Theorem 1.42 [37] Let xnnN form a q-basis for the space X, and the system ynnN is p- close to xnnN, where 1p+1q=1. Then the following properties are equivalent:

1. ynnN is complete in X;

2. ynnN is minimal in X;

3. ynnN forms an isomorphic basis to xnnN for X.

Let X1=XCm and ûnnNX1 be some minimal system, and ϑ̂nnNX1=XCm be its biorthogonal system:

ûn=unαn1αnm;ϑ̂n=ϑnβn1βnm.E75

Let J=n1nm be some set of m natural numbers. Suppose

δ=detβniji,j=1,m̅.E76

The following theorem holds true.

Theorem 1.43 [38] Let the system ûnnN form a basis for X1. In order to the system unnNJ, where NJ=N\J, form a basis for X it is necessary and sufficient that the condition δ0 is satisfied. In this case the biorthogonal system to unnNJ is defined by

ϑn=1δϑnϑn1ϑnmβn1βn11βnm1βnmβn1mβnmm.E77

For δ=0 the system unnNJ is not complete and is not minimal in X.

Let X be a Banach space and uknk=1,m̅;nN be some system in X.

Let aikn,i,k=1,m̅,nN, be some complex numbers. Put

An=aikni,k=1,m̅andΔn=detAn,nN.E78

Let us consider the following system in space X

ûkn=i=1maiknuin,k=1,m̅;nN.E79

Theorem 1.44 If the system uknk=1,m̅;nN forms a basis for X and

Δn0,nN,E80

then the system ûknk=1,m̅;nN forms a basis with parentheses for X. If in addition the conditions

supnAn,An1<,supnuknϑkn<,E81

hold, where ϑknk=1,m̅;nNX is biorthogonal system to uknk=1,m̅;nN, then the system ûknk=1,m̅;nN forms the usual basis for X.

The following theorem holds true.

Theorem 1.45 The system of eigen and associated vectors of the operator L forms the basis for space Mp,α01C,1<p<,0<α1.

Now, let us consider the basicity of the system y0yi,ni=1,2;nN with a removed function in space Mp,α01.

Theorem 1.46 If from the system of eigen and associated functions of problem (67), (68) y0yi,ni=1,2;nN we eliminate any function y2,n0x, corresponding to a simple eigenvalue, then the new system forms a basis for Mp,α01, 1<p<, 0<α1. And if we eliminate any function y1,n0x from this system, then the obtained system does not form a basis in Mp,α01; moreover, in this case this system is not complete and is not minimal in this space.

Proof. For the eigenfunctions z0zi,ni=1,2;nN of the adjoint problem we have z1,n13=0 for any nN and z2,n130. On the other hand, the eigenvectors of the adjoint operator L are defined by ẑn=znm¯zn13,n=0,1,,. Applying Theorem 1.43 to the system ŷ0ŷi,ni=1,2;nN, we notice that δ=m¯z1,n13=0 for any nN and δ=m¯z2,n130 for any eigenfunction corresponding to a simple eigenvalue, and the statements of the theorem follow from the corresponding statements of Theorem 1.43.

Theorem is proved.

Conflict of interest

The authors declare no conflict of interest.

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Written By

Bilal Bilalov, Sabina Sadigova and Zaur Kasumov

Reviewed: 13 April 2022 Published: 11 June 2022