Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011Β (2011), Article IDΒ 243724, 18 pages
http://dx.doi.org/10.5402/2011/243724
Research Article

On Differential Properties π‘…πœˆ-Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data

Computing Center, Far-Eastern Branch, Russian Academy of Sciences, Djambula Street, 25-260, Khabarovsk 680011, Russia

Received 16 February 2011; Accepted 11 April 2011

Academic Editors: G.Β MisioΕ‚ek and C.Β Zhu

Copyright Β© 2011 Victor Anatolievich Rukavishnikov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the first boundary value problem for second-order differential equation with strong singularity caused by coordinated degeneration of the input data. For this problem, we study the differential properties of the π‘…πœˆ-generalized solution, that is, the fact that it belongs to the space π»π‘˜+22,𝜈+𝛽/2(Ξ©).

1. Introduction

In this paper, we study of differential properties of the Dirichlet problem for elliptic equations possessing strong singularity. A boundary value problem is said to possess strong singularity if its solution 𝑒(π‘₯) does not belong to the Sobolev space π‘Š12(𝐻1) or, in other words, the Dirichlet integral of the solution 𝑒(π‘₯) diverges.

Boundary value problems with strong singularity caused by the singularity in the initial data or by the internal properties of solution are found in the physics of plasma and gas discharge, electrodynamics, nuclear physics, nonlinear optics, and other branches of physics. In particular cases, numerical methods for problems of electrodynamics and quantum mechanics with string singularity were constructed based on separation of singular and components, mesh refinement near singular points, multiplicative extraction of singularities, and so forth (see, e.g., [1–4]).

The notion of an π‘…πœˆ-generalized solution was introduced in [5] for boundary value problems with strong singularity in a solution, that is, for problems in which it is impossible to define a generalized (weak) solution or the generalized solution does not have the desired regularity. Such a new concept of solution led to distinction of two classes of boundary value problems: problems with coordinated and uncoordinated degeneration of input data; it also made it possible to study the existence and uniqueness of solutions as well as its coercivity and differential properties in the weighted Sobolev spaces (see [6–8]).

In [9–11], β„Ž,𝑝 and β„Ž-𝑝 versions of the finite element method were constructed and investigated for a Dirichlet problem with strong singularity of solution.

In the present paper, we consider the first boundary value problem for second-order differential equation with strong singularity caused by coordinated degeneration of the input data. For this problem, we study the differential properties of the π‘…πœˆ-generalized solution, that is, the fact that it belongs to the space π»π‘˜+22,𝜈+𝛽/2(Ξ©).

2. The Basic Designations

Let 𝑅2 denote the two-dimensional Euclidean space with π‘₯=(π‘₯1,π‘₯2). Let Ξ© be a bounded convex domain with boundary πœ•Ξ©, and let Ξ© be the closure of Ξ©. We denote by πœ•Ξ©π‘œ the set of the points of πœ•Ξ©: πœ•Ξ©(𝑖)βˆˆπœ•Ξ©, πœ•Ξ©π‘œ=⋃𝑛𝑖=1πœ•Ξ©(𝑖). Let us assume that boundary πœ•Ξ© is a piecewise smooth andπœ•Ξ©β§΅πœ•Ξ©π‘œβˆˆπΆ2.

Let 𝜌(π‘₯) be a weight function that is infinitely differentiable and positive everywhere, except at the points of πœ•Ξ©π‘œ, with 𝜌(π‘₯) coinciding in some neighborhood of each point πœ•Ξ©(𝑖)(𝑖=1,𝑛) with the distance to πœ•Ξ©(𝑖). Moreover, the derivatives of 𝜌(π‘₯) satisfy the inequality ||||πœ•πœŒ(π‘₯)πœ•π‘₯𝑗|||||||||πœ•β‰€π›Ώ,|𝑖|πœŒπ›Ύ(π‘₯)πœ•π‘₯𝑖11πœ•π‘₯𝑖22|||||β‰€πœŽπœŒπ›Ύβˆ’π‘–(π‘₯),(2.1) where 𝑗=1,2;𝑖=(𝑖1,𝑖2), |𝑖|=𝑖1+𝑖2 are nonnegative integers; 𝛿, 𝜎, 𝛾 is real, 𝛿>1.

We introduce the weighted Sobolev space π»π‘˜2,𝛼(Ξ©) which at fixed integer π‘˜β‰₯0 and real 𝛼>βˆ’1 is updating of the set 𝐢∞(Ξ©) infinitely differentiated in Ξ© function on norm (‖𝑒π‘₯)β€–π»π‘˜2,𝛼(Ξ©)=βŽ›βŽœβŽœβŽξ“||πœ†||β‰€π‘˜ξ€œΞ©πœŒ2(𝛼+|πœ†|βˆ’π‘˜)(||𝐷π‘₯)πœ†||𝑒(π‘₯)2βŽžβŽŸβŽŸβŽ π‘‘π‘₯1/2.(2.2) Seminorms in the this space look like ||||𝑒(π‘₯)𝐻𝑠2,𝛼(Ξ©)=βŽ›βŽœβŽœβŽξ“||πœ†||=π‘ ξ€œΞ©πœŒ2(𝛼+|πœ†|βˆ’π‘˜)(||𝐷π‘₯)πœ†||𝑒(π‘₯)2βŽžβŽŸβŽŸβŽ π‘‘π‘₯1/2,(2.3)

where 𝑠 is some integer, 0β‰€π‘ β‰€π‘˜. As usual, 𝐻02,𝛼(Ξ©)=𝐿2,𝛼(Ξ©).

Let us count function πœ‘(π‘₯) from space π»π‘˜βˆ’1/22,𝛼(πœ•Ξ©) if there is such function Ξ¦(π‘₯) from space π»π‘˜2,𝛼(Ξ©) that ||Ξ¦(π‘₯)πœ•Ξ©=πœ‘(π‘₯),β€–πœ‘(π‘₯)β€–π»π‘˜βˆ’1/22,𝛼(πœ•Ξ©)=infΞ¦||πœ•Ξ©=πœ‘β€–Ξ¦(π‘₯)β€–π»π‘˜2,𝛼(Ξ©).(2.4)

Denote by π»π‘˜βˆž,βˆ’π›Ό(Ξ©,𝐢) the set of functions with the norm satisfying the inequality ‖𝑒(π‘₯)β€–π»π‘˜βˆž,βˆ’π›Ό(Ξ©,𝐢)=max||πœ†||β‰€π‘˜vraimaxβˆ€π‘₯∈Ω||πœŒβˆ’π›Ό+|πœ†|(π‘₯)π·πœ†||𝑒(π‘₯)≀𝐢,(2.5) where 𝐢 is a positive constant independent of 𝑒(π‘₯).

As usual, 𝐻0∞,βˆ’π›Ό(Ξ©,𝐢)=𝐿∞,βˆ’π›Ό(Ξ©,𝐢).

3. Statement of Problem: Definition of the π‘…πœˆ-Generalized Solution

Suppose that the differential equation 𝐿𝑒(π‘₯)β‰‘βˆ’2𝑙,𝑠=1π‘Žπ‘™π‘ πœ•(π‘₯)2𝑒(π‘₯)πœ•π‘₯π‘™πœ•π‘₯𝑠+2𝑙=1π‘Žπ‘™(π‘₯)πœ•π‘’(π‘₯)πœ•π‘₯𝑙+π‘Ž(π‘₯)𝑒(π‘₯)=𝑓(π‘₯),π‘₯∈Ω,(3.1) is given in the domain Ξ©, with the boundary condition 𝑒(π‘₯)=πœ‘(π‘₯),π‘₯βˆˆπœ•Ξ©.(3.2) We suppose that π‘Ž12(π‘₯)=π‘Ž21(π‘₯) and for some real π›½π‘Žπ‘™π‘ (π‘₯)∈𝐻1∞,βˆ’π›½ξ€·Ξ©,𝐢1ξ€Έ(𝑙,𝑠=1,2),(3.3)π‘Žπ‘™(π‘₯)∈𝐿∞,βˆ’(π›½βˆ’1)ξ€·Ξ©,𝐢2ξ€Έ(𝑙=1,2),(3.4)π‘Ž(π‘₯)∈𝐿∞,βˆ’(π›½βˆ’2)ξ€·Ξ©,𝐢3ξ€Έ,(3.5)2𝑙,𝑠=1π‘Žπ‘™π‘ (π‘₯)πœ‰π‘™πœ‰π‘ β‰₯𝐢4πœŒπ›½(π‘₯)2𝑙=1πœ‰2𝑙,(3.6)π‘Ž(π‘₯)β‰₯𝐢5πœŒπ›½βˆ’2(π‘₯)a.e.inΞ©,(3.7)𝑓(π‘₯)∈𝐿2,πœ‡(Ξ©),(3.8)πœ‘(π‘₯)∈𝐻1/22,πœ‡+π›½βˆ’1(πœ•Ξ©),(3.9) where πœ‰1, πœ‰2 are arbitrary real parameters, 𝐢𝑖 are positive constants independent of π‘₯(𝑖=1,5), and πœ‡ is nonnegative.

The boundary value problem (3.1), (3.2) under conditions (3.3)–(3.9) will be called the Dirichlet problem with coordinated degeneration of the input data.

We introduce the bilinear and linear forms π‘Ž(𝑒,𝑣)=2𝑙,𝑠=1ξ€œΞ©ξ‚Έπ‘Žπ‘™π‘ (π‘₯)𝜌2𝜈(π‘₯)πœ•π‘’(π‘₯)πœ•π‘₯π‘ πœ•π‘£(π‘₯)πœ•π‘₯𝑙+π‘Žπ‘™π‘ (π‘₯)πœ•πœŒ2𝜈(π‘₯)πœ•π‘₯π‘™πœ•π‘’(π‘₯)πœ•π‘₯𝑠+𝑣(π‘₯)πœ•π‘Žπ‘™π‘ (π‘₯)πœ•π‘₯π‘™πœŒ2𝜈(π‘₯)πœ•π‘’(π‘₯)πœ•π‘₯𝑠+𝑣(π‘₯)𝑑π‘₯2𝑙=1ξ€œΞ©π‘Žπ‘™(π‘₯)𝜌2𝜈(π‘₯)πœ•π‘’(π‘₯)πœ•π‘₯π‘™ξ€œπ‘£(π‘₯)𝑑π‘₯+Ξ©π‘Ž(π‘₯)𝜌2πœˆξ€œ(π‘₯)𝑒(π‘₯)𝑣(π‘₯)𝑑π‘₯,(𝑓,𝑣)=Ω𝜌2𝜈(π‘₯)𝑓(π‘₯)𝑣(π‘₯)𝑑π‘₯.(3.10)

Definition 3.1. A function π‘’πœˆ(π‘₯) in the space𝐻12,𝜈+𝛽/2(Ξ©) will be called π‘…πœˆ-generalized solution of the Dirichlet problem with coordinated degeneration of the input data if π‘’πœˆ(π‘₯)=πœ‘(π‘₯) almost everywhere in πœ•Ξ© and for all 𝑣(π‘₯) in π‘œπ»12,𝜈+𝛽/2(Ξ©), the identity π‘Ž(π‘’πœˆ,𝑣)=(𝑓,𝑣) holds, where 𝜈 is arbitrary but fixed and satisfies the inequality π›½πœˆβ‰₯πœ‡+2βˆ’1.(3.11)

4. Existence, Uniqueness, and Coercive of the π‘…πœˆ-Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data

Theorem 4.1 (see [6]). Assume that conditions (3.3)–(3.11) are satisfied and inequality 2𝐢1(2𝛿|𝜈|+1)+0.5𝐢2ξ€Έ2<𝐢4𝐢5(4.1) holds. Then there exists a unique π‘…πœˆ-generalized solution π‘’πœˆ(π‘₯) of the Dirichlet problem with coordinated degeneration of the input data in the space 𝐻12,𝜈+𝛽/2(Ξ©) and the following estimate holds: β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻12,𝜈+𝛽/2(Ξ©)≀𝐢6‖𝑓(π‘₯)‖𝐿2,πœ‡(Ξ©)+β€–πœ‘(π‘₯)‖𝐻1/22,πœ‡+π›½βˆ’1(πœ•Ξ©),(4.2) where 𝐢6 is a positive constant independent of π‘’πœˆ(π‘₯), 𝑓(π‘₯), and πœ‘(π‘₯).

Theorem 4.2. Assume that conditions (3.3)–(3.8), (4.1) are satisfied and also πœ‘(π‘₯)∈𝐻3/22,πœ‡+𝛽(πœ•Ξ©),(4.3)π›½πœˆβ‰₯πœ‡+2,(4.4)π›½πœˆ+2>2(4.5) holds. Then π‘…πœˆ-generalized solution π‘’πœˆ(π‘₯) of the Dirichlet problem with coordinated degeneration of the input data belongs to space 𝐻22,𝜈+𝛽/2(Ξ©) and coercivity inequality is valid β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻22,𝜈+𝛽/2(Ξ©)≀𝐢7‖𝑓(π‘₯)‖𝐿2,πœ‡(Ξ©)+β€–πœ‘(π‘₯)‖𝐻3/22,πœ‡+𝛽(πœ•Ξ©),(4.6) where 𝐢7 is a positive constant independent of π‘’πœˆ(π‘₯), 𝑓(π‘₯), and πœ‘(π‘₯).
For case πœ‘(π‘₯)=0, the proof of the Theorem 4.2 resulted in [6, Theorem 3].

5. Differential Properties of an π‘…πœˆ-Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data

Theorem 5.1. Let π‘Ž12(π‘₯)=π‘Ž21(π‘₯) and conditions π‘Žπ‘™π‘ (π‘₯)βˆˆπ»π‘˜+1∞,βˆ’π›½ξ€·Ξ©,𝐢1ξ€Έ(𝑙,𝑠=1,2),(5.1)π‘Žπ‘™(π‘₯)βˆˆπ»π‘˜βˆž,βˆ’(π›½βˆ’1)ξ€·Ξ©,𝐢2ξ€Έ(𝑙=1,2),(5.2)π‘Ž(π‘₯)βˆˆπ»π‘˜βˆž,βˆ’(π›½βˆ’2)ξ€·Ξ©,𝐢3ξ€Έ,(5.3)𝑓(π‘₯)βˆˆπ»π‘˜2,πœ‡(Ξ©),(5.4)πœ‘(π‘₯)βˆˆπ»π‘˜+3/22,πœ‡+𝛽(πœ•Ξ©),(5.5)π›½πœˆ+2>π‘˜+2(5.6) are satisfied at some fixed integer π‘˜β‰₯1 and also (3.6), (3.7), (4.1), and (4.4) hold. Then, the π‘…πœˆ-generalized solution π‘’πœˆ(π‘₯) of the Dirichlet problem with coordinated degeneration of the input data belongs to space π»π‘˜+22,𝜈+𝛽/2(Ξ©) and following estimate holds β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘˜+22,𝜈+𝛽/2(Ξ©)≀𝐢8‖𝑓(π‘₯)β€–π»π‘˜2,πœ‡(Ξ©)+β€–πœ‘(π‘₯)β€–π»π‘˜+3/22,πœ‡+𝛽(πœ•Ξ©),(5.7) where 𝐢8 is a positive constant independent of π‘’πœˆ(π‘₯), 𝑓(π‘₯), and πœ‘(π‘₯).

Proof. At performance of conditions (5.1)–(5.6) of the Theorem 5.1 conditions of the Theorem 4.2 are satisfied, therefore, function π‘’πœˆ(π‘₯) belongs to space 𝐻22,𝜈+𝛽/2(Ξ©).
The proof of the Theorem 5.1 we will lead a method of a mathematical induction to two stages: at the first stage, we will check up validity of the statement for π‘˜=1; at the second stage, we will prove it for π‘˜=π‘š the assumption that it is true at π‘˜=π‘šβˆ’1.

Stage 1. Let conditions (3.6), (3.7), (4.1), and (4.4) be satisfied and also conditions π‘Žπ‘™π‘ (π‘₯)∈𝐻2∞,βˆ’π›½ξ€·Ξ©,𝐢1ξ€Έ(𝑙,𝑠=1,2)(5.11)π‘Žπ‘™(π‘₯)∈𝐻1∞,βˆ’(π›½βˆ’1)ξ€·Ξ©,𝐢2ξ€Έ(𝑙=1,2),(5.21)π‘Ž(π‘₯)∈𝐻1∞,βˆ’(π›½βˆ’2)ξ€·Ξ©,𝐢3ξ€Έ,(5.31)𝑓(π‘₯)∈𝐻12,πœ‡(Ξ©),(5.41)πœ‘(π‘₯)∈𝐻5/22,πœ‡+𝛽(πœ•Ξ©),(5.51)π›½πœˆ+2>3(5.61) holds.
Let us designate π‘’πœˆπ‘—(π‘₯)=πœ•π‘’πœˆ(π‘₯)/πœ•π‘₯𝑗(𝑗=1,2). We will prove that functions π‘’πœˆπ‘—(π‘₯) belong to space 𝐻12,𝜈+𝛽/2(Ξ©). For this purpose we will fix 𝑗 and we will establish the upper bound a square of norm β€–β€–π‘’πœˆπ‘—(β€–β€–π‘₯)2𝐻12,𝜈+𝛽/2(Ξ©)=||πœ†||≀1ξ€œΞ©πœŒ2(𝜈+𝛽/2+|πœ†|βˆ’1)(||𝐷π‘₯)πœ†π‘’πœˆπ‘—(||π‘₯)2𝑑π‘₯.(5.8) We consider the sum |πœ†|≀1||π·πœ†π‘’πœˆ1||(π‘₯)2=||π‘’πœˆ1||(π‘₯)2+||||πœ•π‘’πœˆ1(π‘₯)πœ•π‘₯1||||2+||||πœ•π‘’πœˆ1(π‘₯)πœ•π‘₯2||||2=||||πœ•π‘’πœˆ(π‘₯)πœ•π‘₯1||||2+||||πœ•2π‘’πœˆ(π‘₯)πœ•π‘₯21||||2+||||πœ•2π‘’πœˆ(π‘₯)πœ•π‘₯1πœ•π‘₯2||||2.(5.9) We add to the right part of this equality nonnegative composed |π‘’πœˆ(π‘₯)|2, |πœ•π‘’πœˆ(π‘₯)/πœ•π‘₯2|2, |πœ•2π‘’πœˆ(π‘₯)/πœ•π‘₯22|2, |πœ•2π‘’πœˆ(π‘₯)/πœ•π‘₯2πœ•π‘₯1|2. Then, we will receive an inequality |πœ†|≀1||π·πœ†π‘’πœˆ1||(π‘₯)2≀|𝛼|≀2||π·π›Όπ‘’πœˆ||(π‘₯)2.(5.10) Similarly, the inequality turns out |πœ†|≀1||π·πœ†π‘’πœˆ2||(π‘₯)2≀|𝛼|≀2||π·π›Όπ‘’πœˆ||(π‘₯)2.(5.11) Let’s take advantage of designations |𝛼|=|πœ†|+1 for |πœ†|≀1 and two previous inequalities. Then, from (5.8), we will receive an estimation β€–β€–π‘’πœˆπ‘—β€–β€–2𝐻12,𝜈+𝛽/2(Ξ©)≀|𝛼|≀2ξ€œΞ©πœŒ2(𝜈+𝛽/2+(π‘₯)|𝛼|βˆ’2)||π·π›Όπ‘’πœˆ(||π‘₯)2‖‖𝑒𝑑π‘₯=πœˆβ€–β€–2𝐻22,𝜈+𝛽/2(Ξ©).(5.12) From this estimation (as π‘’πœˆ(π‘₯)∈𝐻22,𝜈+𝛽/2(Ξ©)) limitation of function π‘’πœˆπ‘—(π‘₯) in norm of space 𝐻12,𝜈+𝛽/2(Ξ©) follows, that is, π‘’πœˆπ‘—(π‘₯)∈𝐻12,𝜈+𝛽/2(Ξ©).
We write down integrated identity π‘Ž(π‘’πœˆ,𝑣)=(𝑓,𝑣) in the form of 2𝑙,𝑠=1ξ€œΞ©πœ•ξ€·πœŒ2𝜈(π‘₯)π‘Žπ‘™π‘ ξ€Έ(π‘₯)𝑣(π‘₯)πœ•π‘₯π‘™πœ•π‘’πœˆ(π‘₯)πœ•π‘₯𝑠𝑑π‘₯+2𝑙=1ξ€œΞ©πœŒ2𝜈(π‘₯)π‘Žπ‘™(π‘₯)πœ•π‘’πœˆ(π‘₯)πœ•π‘₯𝑙+ξ€œπ‘£(π‘₯)𝑑π‘₯Ω𝜌2𝜈(π‘₯)π‘Ž(π‘₯)π‘’πœˆ(ξ€œπ‘₯)𝑣(π‘₯)𝑑π‘₯=Ω𝜌2𝜈(π‘₯)𝑓(π‘₯)𝑣(π‘₯)𝑑π‘₯.(5.13) In identity (5.13), we will designate 𝑣(π‘₯)=πœ•π‘£1(π‘₯)/πœ•π‘₯𝑗(𝑗=1,2), where function 𝑣1(π‘₯) belongs to space π‘œπ»22,𝜈+𝛽/2+1(Ξ©).
Let us consider the first composed left part of identity (5.13). We will apply to it twice formula of integration in parts (we will consider that 𝑣(π‘₯)=0 if π‘₯βˆˆπœ•Ξ©) and also the formula of differentiation of product of functions. We will receive equality 2𝑙,𝑠=1ξ€œΞ©πœ•ξ€·πœŒ2πœˆπ‘Žπ‘™π‘ π‘£ξ€Έπœ•π‘₯π‘™πœ•π‘’πœˆπœ•π‘₯π‘ ξ€œπ‘‘π‘₯=Ξ©πœ•πœ•π‘₯π‘—ξƒ©βˆ’2𝑙,𝑠=1𝜌2πœˆπ‘Žπ‘™π‘ πœ•2π‘’πœˆπœ•π‘₯π‘™πœ•π‘₯𝑠𝑣1ξƒͺ+ξ€œπ‘‘π‘₯Ξ©πœ•πœŒ2πœˆπœ•π‘₯𝑗2𝑙,𝑠=1π‘Žπ‘™π‘ πœ•2π‘’πœˆπœ•π‘₯π‘™πœ•π‘₯𝑠𝑣1𝑑π‘₯+2𝑙,𝑠=1ξ€œΞ©πœŒ2πœˆπœ•π‘Žπ‘™π‘ πœ•π‘₯π‘—πœ•2π‘’πœˆπœ•π‘₯π‘™πœ•π‘₯𝑠𝑣1βˆ’π‘‘π‘₯2𝑙,𝑠=1ξ€œΞ©πœ•ξ€·πœŒ2πœˆπ‘Žπ‘™π‘ π‘£1ξ€Έπœ•π‘₯π‘™πœ•π‘’πœˆπ‘—πœ•π‘₯𝑠𝑑π‘₯.(5.14) To the second and the third composed left part also for the right part of identity (5.13), we will apply the formula of integration in parts and the formula of differentiation of product of functions. As a result, we will receive equalities 2𝑙=1ξ€œΞ©πœŒ2πœˆπ‘Žπ‘™πœ•π‘’πœˆπœ•π‘₯π‘™ξ€œπ‘£π‘‘π‘₯=Ξ©πœ•πœ•π‘₯𝑗2𝑙=1𝜌2πœˆπ‘Žπ‘™πœ•π‘’πœˆπœ•π‘₯𝑙𝑣1ξƒͺξ€œπ‘‘π‘₯βˆ’Ξ©πœ•πœŒ2πœˆπœ•π‘₯𝑗2𝑙=1π‘Žπ‘™πœ•π‘’πœˆπœ•π‘₯𝑙𝑣1βˆ’π‘‘π‘₯2𝑙=1ξ€œΞ©πœŒ2πœˆπœ•π‘Žπ‘™πœ•π‘₯π‘—πœ•π‘’πœˆπœ•π‘₯𝑙𝑣1𝑑π‘₯βˆ’2𝑙=1ξ€œΞ©πœŒ2πœˆπ‘Žπ‘™πœ•π‘’πœˆπ‘—πœ•π‘₯𝑙𝑣1𝑑π‘₯,(5.15)ξ€œΞ©πœŒ2𝜈(π‘₯)π‘Ž(π‘₯)π‘’πœˆξ€œ(π‘₯)𝑣(π‘₯)𝑑π‘₯=Ξ©πœ•πœ•π‘₯π‘—ξ€·πœŒ2πœˆπ‘Žπ‘’πœˆπ‘£1ξ€Έξ€œπ‘‘π‘₯βˆ’Ξ©πœ•πœŒ2πœˆπœ•π‘₯π‘—π‘Žπ‘’πœˆπ‘£1βˆ’ξ€œπ‘‘π‘₯Ω𝜌2πœˆπœ•π‘Žπœ•π‘₯π‘—π‘’πœˆπ‘£1ξ€œπ‘‘π‘₯βˆ’Ξ©πœŒ2πœˆπ‘Žπ‘’πœˆπ‘—π‘£1𝑑π‘₯,(5.16)ξ€œΞ©πœŒ2πœˆξ€œπ‘“π‘£π‘‘π‘₯=Ξ©πœ•πœ•π‘₯π‘—ξ€·πœŒ2πœˆπ‘“π‘£1ξ€Έξ€œπ‘‘π‘₯βˆ’Ξ©πœ•πœŒ2πœˆπœ•π‘₯𝑗𝑓𝑣1ξ€œπ‘‘π‘₯βˆ’Ξ©πœŒ2πœˆπœ•π‘“πœ•π‘₯𝑗𝑣1𝑑π‘₯.(5.17) In view of equalities (5.14)–(5.15) and formulas of differentiation of the sum of function after a grouping and removal of identical multipliers for brackets identity (5.13), we will lead to the form βˆ’2𝑙,𝑠=1ξ€œΞ©πœ•ξ€·πœŒ2πœˆπ‘Žπ‘™π‘ π‘£1ξ€Έπœ•π‘₯π‘™πœ•π‘’πœˆπ‘—πœ•π‘₯𝑠𝑑π‘₯βˆ’2𝑙=1ξ€œΞ©πœŒ2πœˆπ‘Žπ‘™πœ•π‘’πœˆπ‘—πœ•π‘₯𝑙𝑣1ξ€œπ‘‘π‘₯βˆ’Ξ©πœŒ2πœˆπ‘Žπ‘’πœˆπ‘—π‘£1+ξ€œπ‘‘π‘₯Ξ©πœ•πœ•π‘₯π‘—ξ€·πœŒ2πœˆξ€·πΏπ‘’πœˆξ€Έπ‘£βˆ’π‘“1ξ€Έξ€œπ‘‘π‘₯βˆ’Ξ©πœ•πœŒ2πœˆπœ•π‘₯π‘—ξ€·πΏπ‘’πœˆξ€Έπ‘£βˆ’π‘“1ξ€œπ‘‘π‘₯=βˆ’Ξ©πœŒ2πœˆξƒ©πœ•π‘“πœ•π‘₯𝑗+2𝑙,𝑠=1πœ•π‘Žπ‘™π‘ πœ•π‘₯π‘—πœ•2π‘’πœˆπœ•π‘₯π‘™πœ•π‘₯π‘ βˆ’2𝑙=1πœ•π‘Žπ‘™πœ•π‘₯π‘—πœ•π‘’πœˆπœ•π‘₯π‘™βˆ’πœ•π‘Žπœ•π‘₯π‘—π‘’πœˆξƒͺ𝑣1𝑑π‘₯.(5.18)Difference πΏπ‘’πœˆ(π‘₯)βˆ’π‘“(π‘₯) is equal to zero almost everywhere on Ξ©. The integrated identity will take the form 2𝑙,𝑠=1ξ€œΞ©πœ•ξ€·πœŒ2πœˆπ‘Žπ‘™π‘ π‘£1ξ€Έπœ•π‘₯π‘™πœ•π‘’πœˆπ‘—πœ•π‘₯𝑠𝑑π‘₯+2𝑙=1ξ€œΞ©πœŒ2πœˆπ‘Žπ‘™πœ•π‘’πœˆπ‘—πœ•π‘₯𝑙𝑣1ξ€œπ‘‘π‘₯+Ω𝜌2πœˆπ‘Žπ‘’πœˆπ‘—π‘£1=ξ€œπ‘‘π‘₯Ω𝜌2πœˆξƒ©πœ•π‘“πœ•π‘₯𝑗+2𝑙,𝑠=1πœ•π‘Žπ‘™π‘ πœ•π‘₯π‘—πœ•2π‘’πœˆπœ•π‘₯π‘™πœ•π‘₯π‘ βˆ’2𝑙=1πœ•π‘Žπ‘™πœ•π‘₯π‘—πœ•π‘’πœˆπœ•π‘₯π‘™βˆ’πœ•π‘Žπœ•π‘₯π‘—π‘’πœˆξƒͺ𝑣1𝑑π‘₯,(5.19) or π‘Ž(π‘’πœˆπ‘—,𝑣1)=(𝐹𝑗,𝑣1), where 𝐹𝑗(π‘₯)=πœ•π‘“(π‘₯)πœ•π‘₯𝑗+2𝑙,𝑠=1πœ•π‘Žπ‘™π‘ (π‘₯)πœ•π‘₯π‘—πœ•2π‘’πœˆ(π‘₯)πœ•π‘₯π‘™πœ•π‘₯π‘ βˆ’2𝑙=1πœ•π‘Žπ‘™(π‘₯)πœ•π‘₯π‘—πœ•π‘’πœˆ(π‘₯)πœ•π‘₯π‘™βˆ’πœ•π‘Ž(π‘₯)πœ•π‘₯π‘—π‘’πœˆ(π‘₯).(5.20)
Suppose that the differential equation 𝐿𝑒(π‘₯)=𝐹𝑗(π‘₯),π‘₯∈Ω(5.21) is given in the domain Ξ©, with the boundary condition 𝑒(π‘₯)=πœ‘π‘—(π‘₯),π‘₯βˆˆπœ•Ξ©,(5.22) where πœ‘π‘—(π‘₯)=πœ•π‘’πœˆ(π‘₯)/πœ•π‘₯𝑗|πœ•Ξ©.
Let function π‘’βˆ—(π‘₯) is the π‘…πœˆ-generalized solution of a boundary value problem (5.21), (5.22). Then, for any function π‘£βˆ—(π‘₯)βˆˆπ‘œπ»12,𝜈+𝛽/2(Ξ©) the integrated identity 2𝑙,𝑠=1ξ€œΞ©πœ•ξ€·πœŒ2πœˆπ‘Žπ‘™π‘ π‘£βˆ—ξ€Έπœ•π‘₯π‘™πœ•π‘’βˆ—πœ•π‘₯𝑠𝑑π‘₯+2𝑙=1ξ€œΞ©πœŒ2πœˆπ‘Žπ‘™πœ•π‘’βˆ—πœ•π‘₯π‘™π‘£βˆ—ξ€œπ‘‘π‘₯+Ω𝜌2πœˆπ‘Žπ‘’βˆ—π‘£βˆ—ξ€œπ‘‘π‘₯=Ω𝜌2πœˆπΉπ‘—π‘£βˆ—π‘‘π‘₯(30βˆ—) holds.
Let us prove that function π‘’βˆ—(π‘₯) belongs to space 𝐻22,𝜈+𝛽/2(Ξ©). For this purpose, we will estimate the right parts of (5.21) and a boundary condition (5.22) in norms of corresponding spaces.
First we will prove that the right part of the differential equation (5.21) belongs to space 𝐿2,πœ‡(Ξ©). For this purpose, we will estimate composed in expression 𝐹𝑗(π‘₯). The first composed we will estimate in norm of space 𝐿2,πœ‡(Ξ©)β€–β€–β€–πœ•π‘“(π‘₯)πœ•π‘₯𝑗‖‖‖𝐿2,πœ‡(Ξ©)≀‖𝑓(π‘₯)‖𝐻12,πœ‡(Ξ©).(5.23) The second, the third, and the fourth composed at performance conditions 5.11–5.31 we will estimate in norm of space 𝐿2,πœˆβˆ’π›½/2(Ξ©)β€–β€–β€–β€–2𝑙,𝑠=1πœ•π‘Žπ‘™π‘ (π‘₯)πœ•π‘₯π‘—πœ•2π‘’πœˆ(π‘₯)πœ•π‘₯π‘™πœ•π‘₯𝑠‖‖‖‖𝐿2,πœˆβˆ’π›½/2(Ξ©)≀2𝐢1β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻22,𝜈+𝛽/2βˆ’1(Ξ©),β€–β€–β€–β€–2𝑠=1πœ•π‘Žπ‘ (π‘₯)πœ•π‘₯π‘—πœ•π‘’πœˆ(π‘₯)πœ•π‘₯𝑠‖‖‖‖𝐿2,πœˆβˆ’π›½/2(Ξ©)β‰€βˆš2𝐢2β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻12,𝜈+𝛽/2βˆ’2(Ξ©),β€–β€–β€–πœ•π‘Ž(π‘₯)πœ•π‘₯π‘—π‘’πœˆβ€–β€–β€–(π‘₯)𝐿2,πœˆβˆ’π›½/2(Ξ©)≀𝐢3β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐿2,𝜈+𝛽/2βˆ’3(Ξ©).(5.24) The received estimations are fair for all values of parameter 𝜈 which satisfy condition (4.4). Therefore, they will be fair and for 𝜈=πœ‡+𝛽/2, whence πœˆβˆ’π›½/2=πœ‡. From this fact and four previous estimations, we will receive inequalities ‖‖𝐹𝑗‖‖(π‘₯)𝐿2,πœ‡(Ξ©)β‰€β€–β€–β€–πœ•π‘“(π‘₯)πœ•π‘₯𝑗‖‖‖𝐿2,πœ‡(Ξ©)+β€–β€–β€–β€–2𝑙,𝑠=1πœ•π‘Žπ‘™π‘ (π‘₯)πœ•π‘₯π‘—πœ•2π‘’πœˆ(π‘₯)πœ•π‘₯π‘™πœ•π‘₯𝑠‖‖‖‖𝐿2,πœ‡(Ξ©)+β€–β€–β€–β€–2𝑠=1πœ•π‘Žπ‘ (π‘₯)πœ•π‘₯π‘—πœ•π‘’πœˆ(π‘₯)πœ•π‘₯𝑠‖‖‖‖𝐿2,πœ‡(Ξ©)+β€–β€–β€–πœ•π‘Ž(π‘₯)πœ•π‘₯π‘—π‘’πœˆβ€–β€–β€–(π‘₯)𝐿2,πœ‡(Ξ©)≀‖𝑓(π‘₯)‖𝐻12,πœ‡(Ξ©)+2𝐢1β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻22,𝜈+𝛽/2βˆ’1(Ξ©)+√2𝐢2β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻12,𝜈+𝛽/2βˆ’2(Ξ©)+𝐢3β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐿2,𝜈+𝛽/2βˆ’3(Ξ©).(5.25) Let us strengthen this inequality, and we will receive an estimation ‖‖𝐹𝑗‖‖(π‘₯)𝐿2,πœ‡(Ξ©)≀‖‖𝑓(π‘₯)𝐻12,πœ‡(Ξ©)+𝐢9β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻22,𝜈+𝛽/2βˆ’1(Ξ©),(5.26) where 𝐢9=max{2𝐢1;√2𝐢2;𝐢3}.
The first composed the right part of received estimation is limited under condition of 5.41. Let us prove that second composed too is limited. For this purpose, we will notice that, by analogy with [6, Theorem 2], the π‘…πœˆ-generalized solution of a boundary value problem (3.1), (3.2) is unique (same) for various values of parameter 𝜈. Therefore, function π‘’πœˆ(π‘₯) will satisfy integrated identity (5.13) at 𝜈 and at πœˆβˆ’1. From here and from the Theorem 4.2, the belonging of function π‘’πœˆ(π‘₯) follows space 𝐻22,𝜈+𝛽/2βˆ’1(Ξ©) and also validity of an estimation β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻22,𝜈+𝛽/2βˆ’1(Ξ©)≀𝐢7‖𝑓(π‘₯)‖𝐿2,πœ‡(Ξ©)+β€–πœ‘(π‘₯)‖𝐻3/22,πœ‡+𝛽(πœ•Ξ©).(5.27) From this estimation at performance of inequalities ‖𝑓(π‘₯)‖𝐿2,πœ‡(Ξ©)≀maxβˆ€π‘₯∈Ω𝜌(π‘₯)‖𝑓(π‘₯)‖𝐻12,πœ‡(Ξ©)β€–πœ‘(π‘₯)‖𝐻3/22,πœ‡+𝛽(πœ•Ξ©)≀maxβˆ€π‘₯∈Ω𝜌(π‘₯)β€–πœ‘(π‘₯)‖𝐻5/22,πœ‡+𝛽(πœ•Ξ©),(5.28) the estimation β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻22,𝜈+𝛽/2βˆ’1(Ξ©)≀𝐢7maxβˆ€π‘₯βˆˆΞ©ξ‚€β€–πœŒ(π‘₯)𝑓(π‘₯)‖𝐻12,πœ‡(Ξ©)+β€–πœ‘(π‘₯)‖𝐻5/22,πœ‡+𝛽(πœ•Ξ©)(5.29) follows. Therefore, function 𝐹𝑗(π‘₯) is limited in norm of space 𝐿2,πœ‡(Ξ©), that is, 𝐹𝑗(π‘₯),∈𝐿2,πœ‡(Ξ©). Except for the right part of the differential equation (5.21) on the basic of inequalities (5.26) and (5.29) the estimation is valid ‖‖𝐹𝑗‖‖(π‘₯)𝐿2,πœ‡(Ξ©)≀𝐢10‖𝑓(π‘₯)‖𝐻12,πœ‡(Ξ©)+β€–πœ‘(π‘₯)‖𝐻5/22,πœ‡+𝛽(πœ•Ξ©),(5.30) where 𝐢10 is a positive constant independent of 𝐹𝑗(π‘₯), 𝑓(π‘₯) and πœ‘(π‘₯).
The right part πœ‘π‘—(π‘₯) of a boundary condition (5.22) belongs to space 𝐻3/22,πœ‡+𝛽(πœ•Ξ©). Validity of this fact under condition 5.51 follows from inequality β€–β€–πœ‘π‘—β€–β€–(π‘₯)𝐻3/22,πœ‡+𝛽(πœ•Ξ©)β‰€β€–πœ‘(π‘₯)‖𝐻5/22,πœ‡+𝛽(πœ•Ξ©).(5.31) This inequality is obvious if to use definition of norm in space π»π‘˜+1/22,𝛼(πœ•Ξ©).
Therefore, at performance of conditions 5.11–5.31 and 5.61 conditions (3.3)–(3.5) and (4.5) are carried out automatically.
Thus, for a boundary value problem (5.21), (5.22) conditions of the Theorem 4.2 according to which function π‘’βˆ—(π‘₯) belongs to space 𝐻22,𝜈+𝛽/2(Ξ©) are satisfied all and the estimation β€–π‘’βˆ—β€–(π‘₯)𝐻22,𝜈+𝛽/2(Ξ©)≀𝐢7‖‖𝐹𝑗‖‖(π‘₯)𝐿2,πœ‡(Ξ©)+β€–β€–πœ‘π‘—β€–β€–(π‘₯)𝐻3/22,πœ‡+𝛽(πœ•Ξ©)(5.32) is fair. Besides, under the Theorem 4.1 by virtue of belonging πœ‘π‘—(π‘₯)∈𝐻1/22,πœ‡+π›½βˆ’1(πœ•Ξ©) it is received that function π‘’βˆ—(π‘₯) (the π‘…πœˆ-generalized solution of a boundary value problem (5.21), (5.22)) exists and unique in space 𝐻12,𝜈+𝛽/2(Ξ©).
Let us specify that the integrated identity (30βˆ—) is fair for all functions π‘£βˆ—(π‘₯) fromβ€‰β€‰β€‰π‘œπ»12,𝜈+𝛽/2(Ξ©), therefore, it will be fair and for functions 𝑣1(π‘₯). On the basic of this remark, identities (5.13) and (30βˆ—), and also uniqueness of the π‘…πœˆ-generalized solution of a boundary value problem (5.21), (5.22) it is received, that π‘’πœˆπ‘—(π‘₯)β‰‘π‘’βˆ—(π‘₯), that is, function π‘’πœˆπ‘—(π‘₯) belongs to space 𝐻22,𝜈+𝛽/2(Ξ©) and for it the inequality (5.32) is fairly.
Let us establish limitation seminorm |π‘’πœˆ(π‘₯)|𝐻32,𝜈+𝛽/2(Ξ©). For this purpose, we will estimate its square ||π‘’πœˆ(||π‘₯)2𝐻32,𝜈+𝛽/2(Ξ©)=||πœ†||=3ξ€œΞ©πœŒ2(𝜈+𝛽/2)(||𝐷π‘₯)πœ†π‘’πœˆ(||π‘₯)2𝑑π‘₯,(5.33) from above on inequality |πœ†|=3||π·πœ†π‘’πœˆ||(π‘₯)2≀|πœ†|=2||||π·πœ†ξ‚΅πœ•π‘’πœˆ(π‘₯)πœ•π‘₯1ξ‚Ά||||2+|πœ†|=2||||π·πœ†ξ‚΅πœ•π‘’πœˆ(π‘₯)πœ•π‘₯2ξ‚Ά||||2.(5.34) We will receive inequality ||π‘’πœˆ||(π‘₯)2𝐻32,𝜈+𝛽/2(Ξ©)≀||π‘’πœˆ1||(π‘₯)2𝐻22,𝜈+𝛽/2(Ξ©)+||π‘’πœˆ2||(π‘₯)2𝐻22,𝜈+𝛽/2(Ξ©).(5.35) From inequality (5.35) and function π‘’πœˆπ‘—(π‘₯) (𝑗=1,2) proved above, a belonging space 𝐻22,𝜈+𝛽/2(Ξ©) is followed with limitation seminorm, therefore, π‘’πœˆ(π‘₯)∈𝐻32,𝜈+𝛽/2(Ξ©).
Let us prove now an estimation (5.7) for π‘˜=1. For this purpose we will consider function 𝜌𝜈+𝛽/2(π‘₯)π‘’πœˆ(π‘₯): it belongs to space 𝐻3(Ξ©). It follows from a belonging of function π‘’πœˆ(π‘₯) to space 𝐻32,𝜈+𝛽/2(Ξ©) and statements β€œA” of a lemma 1 of [6]. From statement β€œB” of the same lemma, the inequality β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻32,𝜈+𝛽/2(Ξ©)≀𝐢11β€–β€–πœŒπœˆ+𝛽/2(π‘₯)π‘’πœˆβ€–β€–(π‘₯)𝐻3(Ξ©)(5.36) follows. 𝐢11 is a positive constant independent of π‘’πœˆ(π‘₯).
The norm for function 𝜌𝜈+𝛽/2(π‘₯)π‘’πœˆ(π‘₯) in Sobolev space 𝐻3(Ξ©) can be entered it is equivalent (see [12, page 380]) β€–β€–πœŒπœˆ+𝛽/2(π‘₯)π‘’πœˆβ€–β€–(π‘₯)𝐻3(Ξ©)=β€–β€–πœŒπœˆ+𝛽/2(π‘₯)π‘’πœˆβ€–β€–(π‘₯)𝐿2(Ξ©)+||𝜌𝜈+𝛽/2(π‘₯)π‘’πœˆ||(π‘₯)𝐻3(Ξ©).(5.37) Let us estimate from above composed the right part (5.37). For the first composed truly an inequality β€–β€–πœŒπœˆ+𝛽/2(π‘₯)π‘’πœˆβ€–β€–(π‘₯)𝐿2(Ξ©)≀maxβˆ€π‘₯∈Ω𝜌3‖‖𝑒(π‘₯)πœˆβ€–β€–(π‘₯)𝐿2,𝜈+𝛽/2βˆ’3(Ξ©).(5.38)
For an estimation of a square of the second composed in the right part (5.37), we will take advantage of definition seminorm in space 𝐻3(Ξ©), formula 𝐷3βˆ‘(𝑔(π‘₯)β‹…β„Ž(π‘₯))=3𝑖=0𝐢𝑖3(𝐷𝑖𝑔(π‘₯))β‹…(𝐷3βˆ’π‘–β„Ž(π‘₯)), algebraic inequality (βˆ‘π‘›π‘ž=1π‘Žπ‘ž)2βˆ‘β‰€π‘›β‹…π‘›π‘ž=1π‘Ž2π‘ž and conditions which derivatives of weight function possess. As a result, we will receive an inequality ||𝜌𝜈+𝛽/2(π‘₯)π‘’πœˆ||(π‘₯)2𝐻3(Ξ©)||𝑒≀4𝜈||(π‘₯)2𝐻32,𝜈+𝛽/2(Ξ©)||𝑒+12𝜎𝜈||(π‘₯)2𝐻22,𝜈+𝛽/2βˆ’1(Ξ©)+12𝜎2||π‘’πœˆ||(π‘₯)2𝐻12,𝜈+𝛽/2βˆ’2(Ξ©)+4𝜎3β€–β€–π‘’πœˆβ€–β€–(π‘₯)2𝐿2,𝜈+𝛽/2βˆ’3(Ξ©).(5.39) From estimations (5.36)–(5.39), the inequality β€–β€–π‘’πœˆβ€–β€–(π‘₯)2𝐻32,𝜈+𝛽/2(Ξ©)≀𝐢212||π‘’πœˆ||(π‘₯)2𝐻32,𝜈+𝛽/2(Ξ©)+𝐢213β€–β€–π‘’πœˆβ€–β€–(π‘₯)2𝐻22,𝜈+𝛽/2βˆ’1(Ξ©)(5.40) follows. 𝐢212=8𝐢211, 𝐢213=2𝐢211β‹…max{maxβˆ€π‘₯∈Ω𝜌3(π‘₯)+4𝜎3;12𝜎2;12𝜎}.
From this estimation if to apply inequality βˆšπ‘Ž2+𝑏2β‰€π‘Ž+𝑏 which is true for nonnegative π‘Ž and 𝑏, we will receive β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻32,𝜈+𝛽/2(Ξ©)≀𝐢12||π‘’πœˆ||(π‘₯)𝐻32,𝜈+𝛽/2(Ξ©)+𝐢13β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻22,𝜈+𝛽/2βˆ’1(Ξ©).(5.41) From estimations (5.29)–(5.31), (5.41), and estimation (5.32) which is written down for π‘’πœˆπ‘—(π‘₯)β‰‘π‘’βˆ—(π‘₯) (𝑗=1,2), we will receive an inequality β€–β€–π‘’πœˆβ€–β€–(π‘₯)𝐻32,𝜈+𝛽/2(Ξ©)≀𝐢8‖𝑓(π‘₯)‖𝐻12,πœ‡(Ξ©)+β€–πœ‘(π‘₯)‖𝐻5/22,πœ‡+𝛽(πœ•Ξ©).(231) So, π‘’πœˆ(π‘₯)∈𝐻32,𝜈+𝛽/2(Ξ©), the estimation (5.7) for π‘˜=1 is carried out. Therefore, the statement of the Theorem 5.1 is true for π‘˜=1.

Stage 2. Let us assume that under conditions (3.6), (3.7), (4.1), and (4.4), and conditions π‘Žπ‘™π‘ (π‘₯)βˆˆπ»π‘šβˆž,βˆ’π›½ξ€·Ξ©,𝐢1ξ€Έ(𝑙,𝑠=1,2),(5.12)π‘Žπ‘™(π‘₯)βˆˆπ»π‘šβˆ’1∞,βˆ’(π›½βˆ’1)ξ€·Ξ©,𝐢2ξ€Έ(𝑙=1,2)(5.22)π‘Ž(π‘₯)βˆˆπ»π‘šβˆ’1∞,βˆ’(π›½βˆ’2)ξ€·Ξ©,𝐢3ξ€Έ,(5.32)𝑓(π‘₯)βˆˆπ»π‘šβˆ’12,πœ‡(Ξ©),(5.42)πœ‘(π‘₯)βˆˆπ»π‘š+1/22,πœ‡+𝛽(πœ•Ξ©),(5.52)π›½πœˆ+2>π‘š+1(5.62) the π‘…πœˆ-generalized solution π‘’πœˆ(π‘₯) of the Dirichlet problem with coordinated degeneration of the input data belongs to space π»π‘š+12,𝜈+𝛽/2(Ξ©) and the estimation β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+12,𝜈+𝛽/2(Ξ©)≀𝐢8‖𝑓(π‘₯)β€–π»π‘šβˆ’12,πœ‡(Ξ©)+β€–πœ‘(π‘₯)β€–π»π‘š+1/22,πœ‡+𝛽(πœ•Ξ©)(5.72) is fair.
At the put-forward assumption of validity of the statement of the Theorem 5.1 for π‘˜=π‘šβˆ’1, under conditions (3.6), (3.7), (4.1), and (4.4) and under conditions π‘Žπ‘™π‘ (π‘₯)βˆˆπ»π‘š+1∞,βˆ’π›½ξ€·Ξ©,𝐢1ξ€Έ(𝑙,𝑠=1,2),(5.13)π‘Žπ‘™(π‘₯)βˆˆπ»π‘šβˆž,βˆ’(π›½βˆ’1)ξ€·Ξ©,𝐢2ξ€Έ(𝑙=1,2),(5.23)π‘Ž(π‘₯)βˆˆπ»π‘šβˆž,βˆ’(π›½βˆ’2)ξ€·Ξ©,𝐢3ξ€Έ,(5.33)𝑓(π‘₯)βˆˆπ»π‘š2,πœ‡(Ξ©),(5.43)πœ‘(π‘₯)βˆˆπ»π‘š+3/22,πœ‡+𝛽(πœ•Ξ©),(5.53)π›½πœˆ+2>π‘š+2,(5.63) we will prove that the π‘…πœˆ-generalized solution π‘’πœˆ(π‘₯) of the Dirichlet problem with coordinated degeneration of the input data belongs to space π»π‘š+22,𝜈+𝛽/2(Ξ©) and the estimation β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+22,𝜈+𝛽/2(Ξ©)≀𝐢8‖𝑓(π‘₯)β€–π»π‘š2,πœ‡(Ξ©)+β€–πœ‘(π‘₯)β€–π»π‘š+3/22,πœ‡+𝛽(πœ•Ξ©)(5.73) is fair.
The plan of the proof is the same as at a Stage 1, therefore, we will consider only the most essential details of the proof.
Obviously at performance of conditions 5.13–5.63 conditions 5.12–5.62 are carried out automatically, that is, π‘’πœˆ(π‘₯)βˆˆπ»π‘š+12,𝜈+𝛽/2(Ξ©) and the estimation 5.12–5.62 is fair.
Again, we will take advantage of designations π‘’πœˆπ‘—(π‘₯)=πœ•π‘’πœˆ(π‘₯)/πœ•π‘₯𝑗 for (𝑗=1,2). To similarly how it has been made at a Stage 1, it is possible to show that function π‘’πœˆπ‘—(π‘₯) at fixed 𝑗 is the π‘…πœˆ-generalized solution of a boundary value problem (5.21), (5.22) and belongs to space π»π‘š2,𝜈+𝛽/2(Ξ©). The last follows from an inequality ||π‘’πœˆπ‘—||(π‘₯)π»π‘š2,𝜈+𝛽/2(Ξ©)≀||π‘’πœˆ||(π‘₯)π»π‘š+12,𝜈+𝛽/2(Ξ©)(5.42) and belongings of function π‘’πœˆ(π‘₯) to space π»π‘š+12,𝜈+𝛽/2(Ξ©).
Let us prove that right part 𝐹𝑗(π‘₯) of the differential equation (5.21) belongs to space π»π‘šβˆ’12,πœ‡(Ξ©). For this purpose, we will estimate composed the right part in equality 𝐹𝑗(π‘₯). The first composed we will estimate in norm of space π»π‘šβˆ’12,πœ‡(Ξ©)β€–β€–β€–πœ•π‘“(π‘₯)πœ•π‘₯π‘—β€–β€–β€–π»π‘šβˆ’12,πœ‡(Ξ©)≀‖𝑓(π‘₯)β€–π»π‘š2,πœ‡(Ξ©).(5.43) The second, the third, and the fourth composed we will estimate in norm of space π»π‘šβˆ’12,πœˆβˆ’π›½/2(Ξ©) with use of conditions 5.13–5.33, formulas π·π›Όβˆ‘(𝑔(π‘₯)β‹…β„Ž(π‘₯))=|𝛼|𝑖=0𝐢𝑖|𝛼|(𝐷𝑖𝑔(π‘₯))β‹…(π·π›Όβˆ’π‘–β„Ž(π‘₯)), and algebraic inequalities (βˆ‘π‘›π‘ž=1π‘Žπ‘ž)2βˆ‘β‰€π‘›β‹…π‘›π‘ž=1π‘Ž2π‘ž. As a result, we will receive inequalities β€–β€–β€–β€–2𝑙,𝑠=1πœ•π‘Žπ‘™π‘ (π‘₯)πœ•π‘₯π‘—πœ•2π‘’πœˆ(π‘₯)πœ•π‘₯π‘™πœ•π‘₯π‘ β€–β€–β€–β€–π»π‘šβˆ’12,πœˆβˆ’π›½/2(Ξ©)βˆšβ‰€2(π‘šβˆ’1)π‘šπΆ1β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+12,𝜈+𝛽/2βˆ’1(Ξ©),β€–β€–β€–β€–2𝑠=1πœ•π‘Žπ‘ (π‘₯)πœ•π‘₯π‘—πœ•π‘’πœˆ(π‘₯)πœ•π‘₯π‘ β€–β€–β€–β€–π»π‘šβˆ’12,πœˆβˆ’π›½/2(Ξ©)β‰€βˆšβˆš2(π‘šβˆ’1)π‘šπΆ2β€–β€–π‘’πœˆ(β€–β€–π‘₯)π»π‘š2,𝜈+𝛽/2βˆ’2(Ξ©),β€–β€–β€–πœ•π‘Ž(π‘₯)πœ•π‘₯π‘—π‘’πœˆβ€–β€–β€–(π‘₯)π»π‘šβˆ’12,πœˆβˆ’π›½/2(Ξ©)βˆšβ‰€(π‘šβˆ’1)π‘šπΆ3β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘šβˆ’12,𝜈+𝛽/2βˆ’3(Ξ©).(5.44) The received estimations are fair for all values of parameter 𝜈 which satisfy to condition (4.4). Therefore, they will be fair and for 𝜈=πœ‡+𝛽/2, hence πœˆβˆ’π›½/2=πœ‡. From this fact and the four previous estimations, we will receive inequalities ‖‖𝐹𝑗‖‖(π‘₯)π»π‘šβˆ’12,πœ‡(Ξ©)β‰€β€–β€–β€–πœ•π‘“(π‘₯)πœ•π‘₯π‘—β€–β€–β€–π»π‘šβˆ’12,πœ‡(Ξ©)+β€–β€–β€–β€–2𝑙,𝑠=1πœ•π‘Žπ‘™π‘ (π‘₯)πœ•π‘₯π‘—πœ•2π‘’πœˆ(π‘₯)πœ•π‘₯π‘™πœ•π‘₯π‘ β€–β€–β€–β€–π»π‘šβˆ’12,πœ‡(Ξ©)+β€–β€–β€–β€–2𝑠=1πœ•π‘Žπ‘ (π‘₯)πœ•π‘₯π‘—πœ•π‘’πœˆ(π‘₯)πœ•π‘₯π‘ β€–β€–β€–β€–π»π‘šβˆ’12,πœ‡(Ξ©)+β€–β€–β€–πœ•π‘Ž(π‘₯)πœ•π‘₯π‘—π‘’πœˆβ€–β€–β€–(π‘₯)π»π‘šβˆ’12,πœ‡(Ξ©)≀‖𝑓(π‘₯)β€–π»π‘š2,πœ‡(Ξ©)√+(π‘šβˆ’1)ξ‚†π‘šβ‹…max2𝐢1;√2𝐢2;𝐢3ξ‚‡Γ—ξ‚€β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+12,𝜈+𝛽/2βˆ’1(Ξ©)+β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š2,𝜈+𝛽/2βˆ’2(Ξ©)+β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘šβˆ’12,𝜈+𝛽/2βˆ’3(Ξ©).(5.45) To strengthen this estimation, we will receive an inequality ‖‖𝐹𝑗‖‖(π‘₯)π»π‘šβˆ’12,πœ‡(Ξ©)≀‖𝑓(π‘₯)β€–π»π‘š2,πœ‡(Ξ©)√+(π‘šβˆ’1)π‘šβ‹…πΆ9β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+12,𝜈+𝛽/2βˆ’1(Ξ©).(5.46) The first composed the right part of the received inequality is limited at performance of a condition 5.43. Limitation of the second composed follows from a belonging of function π‘’πœˆ(π‘₯) to space π»π‘š+12,𝜈+𝛽/2βˆ’1(Ξ©) (under the Theorem 4.2 and to integrated identity (5.13) which is fair for function π‘’πœˆ(π‘₯) at 𝜈 and at πœˆβˆ’1). Thus, the estimation 5.72 takes the form β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+12,𝜈+𝛽/2βˆ’1(Ξ©)≀𝐢8‖𝑓(π‘₯)β€–π»π‘šβˆ’12,πœ‡(Ξ©)+β€–πœ‘(π‘₯)β€–π»π‘š+1/22,πœ‡+𝛽(πœ•Ξ©).(5.47) From here by virtue of validity of inequalities ‖𝑓(π‘₯)β€–π»π‘šβˆ’12,πœ‡(Ξ©)≀maxβˆ€π‘₯∈Ω𝜌(π‘₯)‖𝑓(π‘₯)β€–π»π‘š2,πœ‡(Ξ©),β€–πœ‘(π‘₯)β€–π»π‘š+1/22,πœ‡+𝛽(πœ•Ξ©)≀maxβˆ€π‘₯∈Ω𝜌(π‘₯)β€–πœ‘(π‘₯)β€–π»π‘š+3/22,πœ‡+𝛽(πœ•Ξ©),(5.48) the estimation β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+12,𝜈+𝛽/2βˆ’1(Ξ©)≀𝐢8maxβˆ€π‘₯βˆˆΞ©ξ‚€β€–πœŒ(π‘₯)𝑓(π‘₯)β€–π»π‘š2,πœ‡(Ξ©)+β€–πœ‘(π‘₯)β€–π»π‘š+3/22,πœ‡+𝛽(πœ•Ξ©)(5.49) follows. Therefore, function 𝐹𝑗(π‘₯) is limited in norm of space π»π‘šβˆ’12,πœ‡(Ξ©), that is, 𝐹𝑗(π‘₯)βˆˆπ»π‘šβˆ’12,πœ‡(Ξ©). Besides, for the right part of the differential equation (5.21) on the basic of inequalities (5.46) and (5.49) the estimation ‖‖𝐹𝑗‖‖(π‘₯)π»π‘šβˆ’12,πœ‡(Ξ©)≀𝐢14‖𝑓(π‘₯)β€–π»π‘š2,πœ‡(Ξ©)+β€–πœ‘(π‘₯)β€–π»π‘š+3/22,πœ‡+𝛽(πœ•Ξ©)(5.50) will be true. Constant 𝐢14 is positive and does not depend from 𝐹𝑗(π‘₯), 𝑓(π‘₯), and πœ‘(π‘₯).
Right part πœ‘π‘—(π‘₯) of a boundary conditions (5.22) belongs to space π»π‘š+1/22,πœ‡+𝛽(πœ•Ξ©). This fact at performance of a condition 5.42 directly follows from an inequality β€–β€–πœ‘π‘—β€–β€–(π‘₯)π»π‘š+1/22,πœ‡+𝛽(πœ•Ξ©)β‰€β€–πœ‘(π‘₯)β€–π»π‘š+3/22,πœ‡+𝛽(πœ•Ξ©).(5.51)
Therefore, for a boundary value problem (5.21), (5.22) conditions at which function π‘’πœˆπ‘—(π‘₯) belongs to space π»π‘š+12,𝜈+𝛽/2(Ξ©) are satisfied all and the estimation β€–β€–π‘’πœˆπ‘—β€–β€–(π‘₯)π»π‘š+12,𝜈+𝛽/2(Ξ©)≀𝐢8‖‖𝐹𝑗‖‖(π‘₯)π»π‘šβˆ’12,πœ‡(Ξ©)+β€–β€–πœ‘π‘—β€–β€–(π‘₯)π»π‘š+1/22,πœ‡+𝛽(πœ•Ξ©)ξ‚Ά(5.52) is fair (according to the assumption of validity of the statement of the Theorem 5.1 at π‘˜=π‘šβˆ’1).
Let us establish limitation seminorm |π‘’πœˆ(π‘₯)|π»π‘š+22,𝜈+𝛽/2(Ξ©). For this purpose, we will estimate its square ||π‘’πœˆ||(π‘₯)2π»π‘š+22,𝜈+𝛽/2(Ξ©)=||πœ†||=π‘š+2ξ€œΞ©πœŒ2(𝜈+𝛽/2)(||𝐷π‘₯)πœ†π‘’πœˆ(||π‘₯)2𝑑π‘₯,(5.53) from above on inequality |πœ†|=π‘š+2||π·πœ†π‘’πœˆ||(π‘₯)2≀|πœ†|=π‘š+1||||π·πœ†ξ‚΅πœ•π‘’πœˆ(π‘₯)πœ•π‘₯1ξ‚Ά||||2+|πœ†|=π‘š+1||||π·πœ†ξ‚΅πœ•π‘’πœˆ(π‘₯)πœ•π‘₯2ξ‚Ά||||2.(5.54) We will receive inequality ||π‘’πœˆ||(π‘₯)2π»π‘š+22,𝜈+𝛽/2(Ξ©)≀||π‘’πœˆ1||(π‘₯)2π»π‘š+12,𝜈+𝛽/2(Ξ©)+||π‘’πœˆ2||(π‘₯)2π»π‘š+12,𝜈+𝛽/2(Ξ©).(5.55) From an inequality (5.55) and the function π‘’πœˆπ‘—(π‘₯) (𝑗=1,2) proved above a belonging space π»π‘š+12,𝜈+𝛽/2(Ξ©) is followed with limitation seminorm, that is, π‘’πœˆ(π‘₯) belongs to space π»π‘š+22,𝜈+𝛽/2(Ξ©).
Let us prove now an estimation 5.73. For this purpose, we will consider function 𝜌𝜈+𝛽/2(π‘₯)π‘’πœˆ(π‘₯). This function belongs to space π»π‘š+2(Ξ©), that follows from a belonging of function π‘’πœˆ(π‘₯) to space π»π‘š+22,𝜈+𝛽/2(Ξ©) and statement β€œA” a lemma 1 of [6]. From statement β€œB” of the same lemma the inequality β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+22,𝜈+𝛽/2(Ξ©)≀𝐢15β€–β€–πœŒπœˆ+𝛽/2(π‘₯)π‘’πœˆβ€–β€–(π‘₯)π»π‘š+2(Ξ©)(5.56) follows. Constant 𝐢15 is positive and does not depend from π‘’πœˆ(π‘₯).
The norm of function 𝜌𝜈+𝛽/2(π‘₯)π‘’πœˆ(π‘₯) in Sobolev space π»π‘š+2(Ξ©) can be determined the formula β€–β€–πœŒπœˆ+𝛽/2(π‘₯)π‘’πœˆβ€–β€–(π‘₯)π»π‘š+2(Ξ©)=β€–β€–πœŒπœˆ+𝛽/2(π‘₯)π‘’πœˆβ€–β€–(π‘₯)𝐿2(Ξ©)+||𝜌𝜈+𝛽/2π‘’πœˆ||(π‘₯)π»π‘š+2(Ξ©).(5.57) Let us estimate from above composed which enter into the right part of equality (5.57). For the first composed the estimation β€–β€–πœŒπœˆ+𝛽/2(π‘₯)π‘’πœˆβ€–β€–(π‘₯)𝐿2(Ξ©)≀maxβˆ€π‘₯βˆˆΞ©πœŒπ‘š+2‖‖𝑒(π‘₯)πœˆβ€–β€–(π‘₯)𝐿2,𝜈+𝛽/2βˆ’π‘šβˆ’2(Ξ©)(5.58) is fair. For an estimation of a square of the second composed in the right part (5.57), we will take advantage of definition of seminorm in space π»π‘š+2(Ξ©), the formula π·π‘š+2βˆ‘(𝑔(π‘₯)β‹…β„Ž(π‘₯))=π‘š+2𝑖=0πΆπ‘–π‘š+2(𝐷𝑖𝑔(π‘₯))β‹…(π·π‘š+2βˆ’π‘–β„Ž(π‘₯)), an algebraic inequality (βˆ‘π‘›π‘ž=1π‘Žπ‘ž)2βˆ‘β‰€π‘›β‹…π‘›π‘ž=1π‘Ž2π‘ž, and conditions which derivatives of weight function possess. The inequality ||𝜌𝜈+𝛽/2(π‘₯)π‘’πœˆ||(π‘₯)2π»π‘š+2(Ξ©)||𝑒≀(π‘š+3)𝜈||(π‘₯)2π»π‘š+22,𝜈+𝛽/2(Ξ©)||𝑒+(π‘š+3)(π‘š+2)𝜎𝜈||(π‘₯)2π»π‘š+12,𝜈+𝛽/2βˆ’1(Ξ©)+(π‘š+3)(π‘š+2)𝜎2Γ—||π‘’πœˆ||(π‘₯)2π»π‘š2,𝜈+𝛽/2βˆ’2(Ξ©)+β‹―+(π‘š+3)πœŽπ‘š+2β€–β€–π‘’πœˆβ€–β€–(π‘₯)2𝐿2,𝜈+𝛽/2βˆ’π‘šβˆ’2(Ξ©)(5.59) will be as a result received. From inequalities (5.56)–(5.59), the estimation β€–β€–π‘’πœˆβ€–β€–(π‘₯)2π»π‘š+22,𝜈+𝛽/2(Ξ©)≀𝐢216||π‘’πœˆ||(π‘₯)2π»π‘š+22,𝜈+𝛽/2(Ξ©)+𝐢217β€–π‘’πœˆ(π‘₯)β€–2π»π‘š+12,𝜈+𝛽/2βˆ’1(Ξ©)(5.60) follows 𝐢216=2(π‘š+3)𝐢215,𝐢217=2𝐢215ξ‚»maxmaxβˆ€π‘₯βˆˆΞ©πœŒπ‘š+2(π‘₯)+(π‘š+3)πœŽπ‘š+2;(π‘š+3)(π‘š+2)πœŽπ‘š+1ξ‚Ό.;β‹―;(π‘š+3)(π‘š+2)𝜎(5.61) From it on an inequality βˆšπ‘Ž2+𝑏2β‰€π‘Ž+𝑏 which is fair for nonnegative values π‘Ž and 𝑏, the inequality β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+22,𝜈+𝛽/2(Ξ©)≀𝐢16||π‘’πœˆ||(π‘₯)π»π‘š+22,𝜈+𝛽/2(Ξ©)+𝐢17β€–β€–π‘’πœˆβ€–β€–(π‘₯)π»π‘š+12,𝜈+(𝛽/2)βˆ’1(Ξ©)(5.62) follows. From estimation (5.49)–(5.52), (5.62), the proved inequality 5.73 directly follows. Therefore, for π‘˜=π‘š the statement of the Theorem 5.1 is proved.
So, the statement of the Theorem 5.1 is proved for π‘˜=1 and also for π‘˜=π‘š in the assumption at which it is true for π‘˜=π‘šβˆ’1. From these facts on the basic of a method of a mathematical induction validity of the statement of the Theorem 5.1 for any natural value π‘˜ follows. The Theorem 5.1 is proved.

Acknowledgments

The research of the authors was supported by the Russian Foundation of Basic Research under Grant no. 10-01-00060 and by Far-Eastern Branch, Russian Academy of Sciences, Project no. 09-II-CO-01-001.

References

  1. F. Assous, P. Ciarlet Jr., and J. SegrΓ©, β€œNumerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method,” Journal of Computational Physics, vol. 161, no. 1, pp. 218–249, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. M. Costabel, M. Dauge, and C. Schwab, β€œExponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains,” Mathematical Models & Methods in Applied Sciences, vol. 15, no. 4, pp. 575–622, 2005. View at Publisher Β· View at Google Scholar
  3. D. Arroyo, A. Bespalov, and N. Heuer, β€œOn the finite element method for elliptic problems with degenerate and singular coefficients,” Mathematics of Computation, vol. 76, no. 258, pp. 509–537, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. H. Li and V. Nistor, β€œAnalysis of a modified SchrΓΆdinger operator in 2D: regularity, index, and FEM,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 320–338, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. V. A. Rukavishnikov, β€œOn differentiability properties of an RΞ½-generalized solution of the Dirichlet problem,” Soviet mathematicsβ€”Doklady, vol. 40, no. 3, pp. 653–655, 1990. View at Google Scholar
  6. V. A. Rukavishnikov and A. G. Ereklintsev, β€œOn the coercivity of the RΞ½-generalized solution of the first boundary value problem with coordinated degeneration of the input data,” Journal of Difference Equations, vol. 41, no. 12, pp. 1757–1767, 2005. View at Google Scholar
  7. V. A. Rukavishnikov and E. V. Kuznetsova, β€œCoercive estimate for a boundary value problem with noncoordinated degeneration of the input data,” Journal of Difference Equations, vol. 43, no. 4, pp. 550–560, 2007. View at Google Scholar
  8. V. A. Rukavishnikov and E. V. Kuznetsova, β€œThe RΞ½-generalized solution with a singularity of a boundary value problem belongs to the space W2,Ξ½+Ξ²/2+k+1k+2(Ξ©,Ξ΄),” Journal of Difference Equations, vol. 45, no. 6, pp. 913–917, 2009. View at Google Scholar
  9. V. A. Rukavishnikov, β€œMethods of numerical analysis for boundary value problems with strong singularity,” Russian Journal of Numerical Analysis and Mathematical Modelling, vol. 24, no. 6, pp. 565–590, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. V. A. Rukavishnikov and H. I. Rukavishnikova, β€œThe finite element method for a boundary value problem with strong singularity,” Journal of Computational and Applied Mathematics, vol. 234, no. 9, pp. 2870–2882, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. V. A. Rukavishnikov and E. V. Kuznetsova, β€œA finite element method scheme for boundary value problems with noncoordinated degeneration of input data,” Numerical Analysis and Applications, vol. 2, no. 3, pp. 250–259, 2009. View at Publisher Β· View at Google Scholar
  12. S. M. Nikolsky, Approximating Functions of Multiple Variables and Embedding Theorems, Nauka, Moscow, Russia, 1977.