# On Differential PropertiesRν-Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data

Authors: Victor Anatolievich Rukavishnikov
Journal: ISRN Mathematical Analysis
 (2011)

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## 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 theRν-generalized solution, that is, the fact that it belongs to the space H2,ν+β/2k+2(Ω).


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## Body

## 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 solutionu(x) does not belong to the Sobolev space W21(H1) or, in other words, the Dirichlet integral of the solution u(x) 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 anRν-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], h,p and h-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 theRν-generalized solution, that is, the fact that it belongs to the space H2,ν+β/2k+2(Ω).


## 2. The Basic Designations

LetR2 denote the two-dimensional Euclidean space with x=(x1,x2). Let Ω be a bounded convex domain with boundary ∂Ω, and let Ω¯ be the closure of Ω. We denote by ∂Ωo the set of the points of ∂Ω: ∂Ω(i)∈∂Ω, ∂Ωo=⋃i=1n∂Ω(i). Let us assume that boundary ∂Ω is a piecewise smooth and∂Ω∖∂Ωo∈C2.Letρ(x) be a weight function that is infinitely differentiable and positive everywhere, except at the points of ∂Ωo, with ρ(x) coinciding in some neighborhood of each point ∂Ω(i)(i=1,n¯) with the distance to ∂Ω(i). Moreover, the derivatives of ρ(x) satisfy the inequality (2.1)|∂ρ(x)∂xj|≤δ,|∂|i|ργ(x)∂x1i1∂x2i2|≤σργ-i(x),
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 spaceH2,αk(Ω) which at fixed integer k≥0 and real α>-1 is updating of the set C∞(Ω¯) infinitely differentiated in Ω¯ function on norm (2.2)‖u(x)‖H2,αk(Ω)=(∑|λ|≤k∫Ωρ2(α+|λ|-k)(x)|Dλu(x)|2dx)1/2.
Seminorms in the this space look like (2.3)|u(x)|H2,αs(Ω)=(∑|λ|=s∫Ωρ2(α+|λ|-k)(x)|Dλu(x)|2dx)1/2,wheres is some integer, 0≤s≤k. As usual, H2,α0(Ω)=L2,α(Ω).Let us count functionφ(x) from space H2,αk-1/2(∂Ω) if there is such function Φ(x) from space H2,αk(Ω) that (2.4)Φ(x)|∂Ω=φ(x),‖φ(x)‖H2,αk-1/2(∂Ω)=inf⁡Φ|∂Ω=φ‖Φ(x)‖H2,αk(Ω).Denote byH∞,-αk(Ω,C) the set of functions with the norm satisfying the inequality (2.5)‖u(x)‖H∞,-αk(Ω,C)=max⁡|λ|≤kvraimax∀x∈Ω|ρ-α+|λ|(x)Dλu(x)|≤C,
where C is a positive constant independent of u(x).As usual,H∞,-α0(Ω,C)=L∞,-α(Ω,C).


## 3. Statement of Problem: Definition of theRν-Generalized Solution

Suppose that the differential equation(3.1)Lu(x)≡-∑l,s=12als(x)∂2u(x)∂xl∂xs+∑l=12al(x)∂u(x)∂xl+a(x)u(x)=f(x),x∈Ω,
is given in the domain Ω, with the boundary condition (3.2)u(x)=φ(x),x∈∂Ω.
We suppose that a12(x)=a21(x) and for some real β(3.3)als(x)∈H∞,-β1(Ω,C1)(l,s=1,2),(3.4)al(x)∈L∞,-(β-1)(Ω,C2)(l=1,2),(3.5)a(x)∈L∞,-(β-2)(Ω,C3),(3.6)∑l,s=12als(x)ξlξs≥C4ρβ(x)∑l=12ξl2,(3.7)a(x)≥C5ρβ-2(x)a.e.inΩ,(3.8)f(x)∈L2,μ(Ω),(3.9)φ(x)∈H2,μ+β-11/2(∂Ω),
where ξ1, ξ2 are arbitrary real parameters, Ci are positive constants independent of x(i=1,5¯), and μ 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(3.10)a(u,v)=∑l,s=12∫Ω[als(x)ρ2ν(x)∂u(x)∂xs∂v(x)∂xl+als(x)∂ρ2ν(x)∂xl∂u(x)∂xsv(x)+∂als(x)∂xlρ2ν(x)∂u(x)∂xsv(x)]dx+∑l=12∫Ωal(x)ρ2ν(x)∂u(x)∂xlv(x)dx+∫Ωa(x)ρ2ν(x)u(x)v(x)dx,(f,v)=∫Ωρ2ν(x)f(x)v(x)dx.Definition 3.1.
A functionuν(x) in the spaceH2,ν+β/21(Ω) will be called Rν-generalized solution of the Dirichlet problem with coordinated degeneration of the input data if uν(x)=φ(x) almost everywhere in ∂Ω and for all v(x) in Ho2,ν+β/21(Ω), the identity a(uν,v)=(f,v) holds, where ν is arbitrary but fixed and satisfies the inequality
(3.11)ν≥μ+β2-1.


## 4. Existence, Uniqueness, and Coercive of theRν-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
(4.1)2(C1(2δ|ν|+1)+0.5C2)2<C4C5
holds. Then there exists a unique Rν-generalized solution uν(x) of the Dirichlet problem with coordinated degeneration of the input data in the space H2,ν+β/21(Ω) and the following estimate holds:
(4.2)‖uν(x)‖H2,ν+β/21(Ω)≤C6(‖f(x)‖L2,μ(Ω)+‖φ(x)‖H2,μ+β-11/2(∂Ω)),
where C6 is a positive constant independent of uν(x), f(x), and φ(x).Theorem 4.2.
Assume that conditions (3.3)–(3.8), (4.1) are satisfied and also
(4.3)φ(x)∈H2,μ+β3/2(∂Ω),(4.4)ν≥μ+β2,(4.5)ν+β2>2
holds. Then Rν-generalized solution uν(x) of the Dirichlet problem with coordinated degeneration of the input data belongs to space H2,ν+β/22(Ω) and coercivity inequality is valid
(4.6)‖uν(x)‖H2,ν+β/22(Ω)≤C7(‖f(x)‖L2,μ(Ω)+‖φ(x)‖H2,μ+β3/2(∂Ω)),
where C7 is a positive constant independent of uν(x), f(x), and φ(x).
For caseφ(x)=0, the proof of the Theorem 4.2 resulted in [6, Theorem 3].


## 5. Differential Properties of anRν-Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data

Theorem 5.1.
Leta12(x)=a21(x) and conditions
(5.1)als(x)∈H∞,-βk+1(Ω,C1)(l,s=1,2),(5.2)al(x)∈H∞,-(β-1)k(Ω,C2)(l=1,2),(5.3)a(x)∈H∞,-(β-2)k(Ω,C3),(5.4)f(x)∈H2,μk(Ω),(5.5)φ(x)∈H2,μ+βk+3/2(∂Ω),(5.6)ν+β2>k+2
are satisfied at some fixed integer k≥1 and also (3.6), (3.7), (4.1), and (4.4) hold. Then, the Rν-generalized solution uν(x) of the Dirichlet problem with coordinated degeneration of the input data belongs to space H2,ν+β/2k+2(Ω) and following estimate holds
(5.7)‖uν(x)‖H2,ν+β/2k+2(Ω)≤C8(‖f(x)‖H2,μk(Ω)+‖φ(x)‖H2,μ+βk+3/2(∂Ω)),
where C8 is a positive constant independent of uν(x), f(x), and φ(x).Proof.
At performance of conditions (5.1)–(5.6) of the Theorem 5.1 conditions of the Theorem 4.2 are satisfied, therefore, function uν(x) belongs to space H2,ν+β/22(Ω).
The proof of the Theorem5.1 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 k=1; at the second stage, we will prove it for k=m the assumption that it is true at k=m-1.Stage 1.
Let conditions (3.6), (3.7), (4.1), and (4.4) be satisfied and also conditions
(5.11)als(x)∈H∞,-β2(Ω,C1)(l,s=1,2)(5.21)al(x)∈H∞,-(β-1)1(Ω,C2)(l=1,2),(5.31)a(x)∈H∞,-(β-2)1(Ω,C3),(5.41)f(x)∈H2,μ1(Ω),(5.51)φ(x)∈H2,μ+β5/2(∂Ω),(5.61)ν+β2>3
holds.
Let us designateuνj(x)=∂uν(x)/∂xj(j=1,2). We will prove that functions uνj(x) belong to space H2,ν+β/21(Ω). For this purpose we will fix j and we will establish the upper bound a square of norm
(5.8)‖uνj(x)‖2H2,ν+β/21(Ω)=∑|λ|≤1∫Ωρ2(ν+β/2+|λ|-1)(x)|Dλuνj(x)|2dx.
We consider the sum
(5.9)∑|λ|≤1|Dλuν1(x)|2=|uν1(x)|2+|∂uν1(x)∂x1|2+|∂uν1(x)∂x2|2=|∂uν(x)∂x1|2+|∂2uν(x)∂x12|2+|∂2uν(x)∂x1∂x2|2.
We add to the right part of this equality nonnegative composed |uν(x)|2, |∂uν(x)/∂x2|2, |∂2uν(x)/∂x22|2, |∂2uν(x)/∂x2∂x1|2. Then, we will receive an inequality
(5.10)∑|λ|≤1|Dλuν1(x)|2≤∑|α|≤2|Dαuν(x)|2.
Similarly, the inequality turns out
(5.11)∑|λ|≤1|Dλuν2(x)|2≤∑|α|≤2|Dαuν(x)|2.
Let’s take advantage of designations |α|=|λ|+1 for |λ|≤1 and two previous inequalities. Then, from (5.8), we will receive an estimation
(5.12)‖uνj‖H2,ν+β/21(Ω)2≤∑|α|≤2∫Ωρ2(ν+β/2+(x)|α|-2)|Dαuν(x)|2dx=‖uν‖H2,ν+β/22(Ω)2.
From this estimation (as uν(x)∈H2,ν+β/22(Ω)) limitation of function uνj(x) in norm of space H2,ν+β/21(Ω) follows, that is, uνj(x)∈H2,ν+β/21(Ω).
We write down integrated identitya(uν,v)=(f,v) in the form of
(5.13)∑l,s=12∫Ω∂(ρ2ν(x)als(x)v(x))∂xl∂uν(x)∂xsdx+∑l=12∫Ωρ2ν(x)al(x)∂uν(x)∂xlv(x)dx+∫Ωρ2ν(x)a(x)uν(x)v(x)dx=∫Ωρ2ν(x)f(x)v(x)dx.
In identity (5.13), we will designate v(x)=∂v1(x)/∂xj(j=1,2), where function v1(x) belongs to space Ho2,ν+β/2+12(Ω).
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(x)=0 if x∈∂Ω) and also the formula of differentiation of product of functions. We will receive equality
(5.14)∑l,s=12∫Ω∂(ρ2νalsv)∂xl∂uν∂xsdx=∫Ω∂∂xj(-∑l,s=12ρ2νals∂2uν∂xl∂xsv1)dx+∫Ω∂ρ2ν∂xj∑l,s=12als∂2uν∂xl∂xsv1dx+∑l,s=12∫Ωρ2ν∂als∂xj∂2uν∂xl∂xsv1dx-∑l,s=12∫Ω∂(ρ2νalsv1)∂xl∂uνj∂xsdx.
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
(5.15)∑l=12∫Ωρ2νal∂uν∂xlvdx=∫Ω∂∂xj(∑l=12ρ2νal∂uν∂xlv1)dx-∫Ω∂ρ2ν∂xj∑l=12al∂uν∂xlv1dx-∑l=12∫Ωρ2ν∂al∂xj∂uν∂xlv1dx-∑l=12∫Ωρ2νal∂uνj∂xlv1dx,(5.16)∫Ωρ2ν(x)a(x)uν(x)v(x)dx=∫Ω∂∂xj(ρ2νauνv1)dx-∫Ω∂ρ2ν∂xjauνv1dx-∫Ωρ2ν∂a∂xjuνv1dx-∫Ωρ2νauνjv1dx,(5.17)∫Ωρ2νfvdx=∫Ω∂∂xj(ρ2νfv1)dx-∫Ω∂ρ2ν∂xjfv1dx-∫Ωρ2ν∂f∂xjv1dx.
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
(5.18)-∑l,s=12∫Ω∂(ρ2νalsv1)∂xl∂uνj∂xsdx-∑l=12∫Ωρ2νal∂uνj∂xlv1dx-∫Ωρ2νauνjv1dx+∫Ω∂∂xj(ρ2ν(Luν-f)v1)dx-∫Ω∂ρ2ν∂xj(Luν-f)v1dx=-∫Ωρ2ν(∂f∂xj+∑l,s=12∂als∂xj∂2uν∂xl∂xs-∑l=12∂al∂xj∂uν∂xl-∂a∂xjuν)v1dx.Difference Luν(x)-f(x) is equal to zero almost everywhere on Ω. The integrated identity will take the form
(5.19)∑l,s=12∫Ω∂(ρ2νalsv1)∂xl∂uνj∂xsdx+∑l=12∫Ωρ2νal∂uνj∂xlv1dx+∫Ωρ2νauνjv1dx=∫Ωρ2ν(∂f∂xj+∑l,s=12∂als∂xj∂2uν∂xl∂xs-∑l=12∂al∂xj∂uν∂xl-∂a∂xjuν)v1dx,
or a(uνj,v1)=(Fj,v1), where
(5.20)Fj(x)=∂f(x)∂xj+∑l,s=12∂als(x)∂xj∂2uν(x)∂xl∂xs-∑l=12∂al(x)∂xj∂uν(x)∂xl-∂a(x)∂xjuν(x).
Suppose that the differential equation(5.21)Lu(x)=Fj(x),x∈Ω
is given in the domain Ω, with the boundary condition
(5.22)u(x)=φj(x),x∈∂Ω,
where φj(x)=∂uν(x)/∂xj|∂Ω.
Let functionu*(x) is the Rν-generalized solution of a boundary value problem (5.21), (5.22). Then, for any function v*(x)∈Ho2,ν+β/21(Ω) the integrated identity
(30*)∑l,s=12∫Ω∂(ρ2νalsv*)∂xl∂u*∂xsdx+∑l=12∫Ωρ2νal∂u*∂xlv*dx+∫Ωρ2νau*v*dx=∫Ωρ2νFjv*dx
holds.
Let us prove that functionu*(x) belongs to space H2,ν+β/22(Ω). 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,μ(Ω). For this purpose, we will estimate composed in expression Fj(x). The first composed we will estimate in norm of space L2,μ(Ω)(5.23)‖∂f(x)∂xj‖L2,μ(Ω)≤‖f(x)‖H2,μ1(Ω).
The second, the third, and the fourth composed at performance conditions 5.11–5.31 we will estimate in norm of space L2,ν-β/2(Ω)(5.24)‖∑l,s=12∂als(x)∂xj∂2uν(x)∂xl∂xs‖L2,ν-β/2(Ω)≤2C1‖uν(x)‖H2,ν+β/2-12(Ω),‖∑s=12∂as(x)∂xj∂uν(x)∂xs‖L2,ν-β/2(Ω)≤2C2‖uν(x)‖H2,ν+β/2-21(Ω),‖∂a(x)∂xjuν(x)‖L2,ν-β/2(Ω)≤C3‖uν(x)‖L2,ν+β/2-3(Ω).
The received estimations are fair for all values of parameter ν which satisfy condition (4.4). Therefore, they will be fair and for ν=μ+β/2, whence ν-β/2=μ. From this fact and four previous estimations, we will receive inequalities
(5.25)‖Fj(x)‖L2,μ(Ω)≤‖∂f(x)∂xj‖L2,μ(Ω)+‖∑l,s=12∂als(x)∂xj∂2uν(x)∂xl∂xs‖L2,μ(Ω)+‖∑s=12∂as(x)∂xj∂uν(x)∂xs‖L2,μ(Ω)+‖∂a(x)∂xjuν(x)‖L2,μ(Ω)≤‖f(x)‖H2,μ1(Ω)+2C1‖uν(x)‖H2,ν+β/2-12(Ω)+2C2‖uν(x)‖H2,ν+β/2-21(Ω)+C3‖uν(x)‖L2,ν+β/2-3(Ω).
Let us strengthen this inequality, and we will receive an estimation
(5.26)‖Fj(x)‖L2,μ(Ω)≤‖f(x)‖H2,μ1(Ω)+C9‖uν(x)‖H2,ν+β/2-12(Ω),
where C9=max{2C1;2C2;C3}.
The first composed the right part of received estimation is limited under condition of5.41. Let us prove that second composed too is limited. For this purpose, we will notice that, by analogy with [6, Theorem 2], the Rν-generalized solution of a boundary value problem (3.1), (3.2) is unique (same) for various values of parameter ν. Therefore, function uν(x) will satisfy integrated identity (5.13) at ν and at ν-1. From here and from the Theorem 4.2, the belonging of function uν(x) follows space H2,ν+β/2-12(Ω) and also validity of an estimation
(5.27)‖uν(x)‖H2,ν+β/2-12(Ω)≤C7(‖f(x)‖L2,μ(Ω)+‖φ(x)‖H2,μ+β3/2(∂Ω)).
From this estimation at performance of inequalities
(5.28)‖f(x)‖L2,μ(Ω)≤max⁡∀x∈Ω¯ρ(x)‖f(x)‖H2,μ1(Ω)∥φ(x)∥H2,μ+β3/2(∂Ω)≤max⁡∀x∈Ω¯ρ(x)∥φ(x)∥H2,μ+β5/2(∂Ω),
the estimation
(5.29)‖uν(x)‖H2,ν+β/2-12(Ω)≤C7max⁡∀x∈Ω¯ρ(x)(‖f(x)‖H2,μ1(Ω)+‖φ(x)‖H2,μ+β5/2(∂Ω))
follows. Therefore, function Fj(x) is limited in norm of space L2,μ(Ω), that is, Fj(x),∈L2,μ(Ω). 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
(5.30)‖Fj(x)‖L2,μ(Ω)≤C10(‖f(x)‖H2,μ1(Ω)+‖φ(x)‖H2,μ+β5/2(∂Ω)),
where C10 is a positive constant independent of Fj(x), f(x) and φ(x).
The right partφj(x) of a boundary condition (5.22) belongs to space H2,μ+β3/2(∂Ω). Validity of this fact under condition 5.51 follows from inequality
(5.31)‖φj(x)‖H2,μ+β3/2(∂Ω)≤‖φ(x)‖H2,μ+β5/2(∂Ω).
This inequality is obvious if to use definition of norm in space H2,αk+1/2(∂Ω).
Therefore, at performance of conditions5.11–5.31 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 u*(x) belongs to space H2,ν+β/22(Ω) are satisfied all and the estimation
(5.32)‖u*(x)‖H2,ν+β/22(Ω)≤C7(‖Fj(x)‖L2,μ(Ω)+‖φj(x)‖H2,μ+β3/2(∂Ω))
is fair. Besides, under the Theorem 4.1 by virtue of belonging φj(x)∈H2,μ+β-11/2(∂Ω) it is received that function u*(x) (the Rν-generalized solution of a boundary value problem (5.21), (5.22)) exists and unique in space H2,ν+β/21(Ω).
Let us specify that the integrated identity(30*) is fair for all functions v*(x) from   Ho2,ν+β/21(Ω), therefore, it will be fair and for functions v1(x). On the basic of this remark, identities (5.13) and (30*), and also uniqueness of the Rν-generalized solution of a boundary value problem (5.21), (5.22) it is received, that uνj(x)≡u*(x), that is, function uνj(x) belongs to space H2,ν+β/22(Ω) and for it the inequality (5.32) is fairly.
Let us establish limitation seminorm|uν(x)|H2,ν+β/23(Ω). For this purpose, we will estimate its square
(5.33)|uν(x)|2H2,ν+β/23(Ω)=∑|λ|=3∫Ωρ2(ν+β/2)(x)|Dλuν(x)|2dx,
from above on inequality
(5.34)∑|λ|=3|Dλuν(x)|2≤∑|λ|=2|Dλ(∂uν(x)∂x1)|2+∑|λ|=2|Dλ(∂uν(x)∂x2)|2.
We will receive inequality
(5.35)|uν(x)|2H2,ν+β/23(Ω)≤|uν1(x)|2H2,ν+β/22(Ω)+|uν2(x)|2H2,ν+β/22(Ω).
From inequality (5.35) and function uνj(x) (j=1,2) proved above, a belonging space H2,ν+β/22(Ω) is followed with limitation seminorm, therefore, uν(x)∈H2,ν+β/23(Ω).
Let us prove now an estimation (5.7) for k=1. For this purpose we will consider function ρν+β/2(x)uν(x): it belongs to space H3(Ω). It follows from a belonging of function uν(x) to space H2,ν+β/23(Ω) and statements “A” of a lemma 1 of [6]. From statement “B” of the same lemma, the inequality
(5.36)‖uν(x)‖H2,ν+β/23(Ω)≤C11‖ρν+β/2(x)uν(x)‖H3(Ω)
follows. C11 is a positive constant independent of uν(x).
The norm for functionρν+β/2(x)uν(x) in Sobolev space H3(Ω) can be entered it is equivalent (see [12, page 380])
(5.37)‖ρν+β/2(x)uν(x)‖H3(Ω)=‖ρν+β/2(x)uν(x)‖L2(Ω)+|ρν+β/2(x)uν(x)|H3(Ω).
Let us estimate from above composed the right part (5.37). For the first composed truly an inequality
(5.38)‖ρν+β/2(x)uν(x)‖L2(Ω)≤max⁡∀x∈Ω¯ρ3(x)‖uν(x)‖L2,ν+β/2-3(Ω).
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(Ω), formula D3(g(x)·h(x))=∑i=03C3i(Dig(x))·(D3-ih(x)), algebraic inequality (∑q=1naq)2≤n·∑q=1naq2 and conditions which derivatives of weight function possess. As a result, we will receive an inequality
(5.39)|ρν+β/2(x)uν(x)|H3(Ω)2≤4|uν(x)|H2,ν+β/23(Ω)2+12σ|uν(x)|H2,ν+β/2-12(Ω)2+12σ2|uν(x)|H2,ν+β/2-21(Ω)2+4σ3‖uν(x)‖L2,ν+β/2-3(Ω).2
From estimations (5.36)–(5.39), the inequality
(5.40)‖uν(x)‖H2,ν+β/23(Ω)2≤C122|uν(x)|H2,ν+β/23(Ω)2+C132‖uν(x)‖H2,ν+β/2-12(Ω)2
follows. C122=8C112, C132=2C112·max{max∀x∈Ω¯ρ3(x)+4σ3;12σ2;12σ}.
From this estimation if to apply inequalitya2+b2≤a+b which is true for nonnegative a and b, we will receive
(5.41)‖uν(x)‖H2,ν+β/23(Ω)≤C12|uν(x)|H2,ν+β/23(Ω)+C13‖uν(x)‖H2,ν+β/2-12(Ω).
From estimations (5.29)–(5.31), (5.41), and estimation (5.32) which is written down for uνj(x)≡u*(x) (j=1,2), we will receive an inequality
(231)‖uν(x)‖H2,ν+β/23(Ω)≤C8(‖f(x)‖H2,μ1(Ω)+‖φ(x)‖H2,μ+β5/2(∂Ω)).
So, uν(x)∈H2,ν+β/23(Ω), the estimation (5.7) for k=1 is carried out. Therefore, the statement of the Theorem 5.1 is true for k=1.Stage 2.
Let us assume that under conditions (3.6), (3.7), (4.1), and (4.4), and conditions
(5.12)als(x)∈H∞,-βm(Ω,C1)(l,s=1,2),(5.22)al(x)∈H∞,-(β-1)m-1(Ω,C2)(l=1,2)(5.32)a(x)∈H∞,-(β-2)m-1(Ω,C3),(5.42)f(x)∈H2,μm-1(Ω),(5.52)φ(x)∈H2,μ+βm+1/2(∂Ω),(5.62)ν+β2>m+1
the Rν-generalized solution uν(x) of the Dirichlet problem with coordinated degeneration of the input data belongs to space H2,ν+β/2m+1(Ω) and the estimation
(5.72)‖uν(x)‖H2,ν+β/2m+1(Ω)≤C8(‖f(x)‖H2,μm-1(Ω)+‖φ(x)‖H2,μ+βm+1/2(∂Ω))
is fair.
At the put-forward assumption of validity of the statement of the Theorem5.1 for k=m-1, under conditions (3.6), (3.7), (4.1), and (4.4) and under conditions
(5.13)als(x)∈H∞,-βm+1(Ω,C1)(l,s=1,2),(5.23)al(x)∈H∞,-(β-1)m(Ω,C2)(l=1,2),(5.33)a(x)∈H∞,-(β-2)m(Ω,C3),(5.43)f(x)∈H2,μm(Ω),(5.53)φ(x)∈H2,μ+βm+3/2(∂Ω),(5.63)ν+β2>m+2,
we will prove that the Rν-generalized solution uν(x) of the Dirichlet problem with coordinated degeneration of the input data belongs to space H2,ν+β/2m+2(Ω) and the estimation
(5.73)‖uν(x)‖H2,ν+β/2m+2(Ω)≤C8(‖f(x)‖H2,μm(Ω)+‖φ(x)‖H2,μ+βm+3/2(∂Ω))
is fair.
The plan of the proof is the same as at a Stage1, therefore, we will consider only the most essential details of the proof.
Obviously at performance of conditions5.13–5.63 conditions 5.12–5.62 are carried out automatically, that is, uν(x)∈H2,ν+β/2m+1(Ω) and the estimation 5.12–5.62 is fair.
Again, we will take advantage of designationsuνj(x)=∂uν(x)/∂xj for (j=1,2). To similarly how it has been made at a Stage 1, it is possible to show that function uνj(x) at fixed j is the Rν-generalized solution of a boundary value problem (5.21), (5.22) and belongs to space H2,ν+β/2m(Ω). The last follows from an inequality
(5.42)|uνj(x)|H2,ν+β/2m(Ω)≤|uν(x)|H2,ν+β/2m+1(Ω)
and belongings of function uν(x) to space H2,ν+β/2m+1(Ω).
Let us prove that right partFj(x) of the differential equation (5.21) belongs to space H2,μm-1(Ω). For this purpose, we will estimate composed the right part in equality Fj(x). The first composed we will estimate in norm of space H2,μm-1(Ω)(5.43)‖∂f(x)∂xj‖H2,μm-1(Ω)≤‖f(x)‖H2,μm(Ω).
The second, the third, and the fourth composed we will estimate in norm of space H2,ν-β/2m-1(Ω) with use of conditions 5.13–5.33, formulas Dα(g(x)·h(x))=∑i=0|α|C|α|i(Dig(x))·(Dα-ih(x)), and algebraic inequalities (∑q=1naq)2≤n·∑q=1naq2. As a result, we will receive inequalities
(5.44)‖∑l,s=12∂als(x)∂xj∂2uν(x)∂xl∂xs‖H2,ν-β/2m-1(Ω)≤2(m-1)mC1‖uν(x)‖H2,ν+β/2-1m+1(Ω),‖∑s=12∂as(x)∂xj∂uν(x)∂xs‖H2,ν-β/2m-1(Ω)≤2(m-1)mC2‖uν(x)‖H2,ν+β/2-2m(Ω),‖∂a(x)∂xjuν(x)‖H2,ν-β/2m-1(Ω)≤(m-1)mC3‖uν(x)‖H2,ν+β/2-3m-1(Ω).
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=μ+β/2, hence ν-β/2=μ. From this fact and the four previous estimations, we will receive inequalities
(5.45)‖Fj(x)‖H2,μm-1(Ω)≤‖∂f(x)∂xj‖H2,μm-1(Ω)+‖∑l,s=12∂als(x)∂xj∂2uν(x)∂xl∂xs‖H2,μm-1(Ω)+‖∑s=12∂as(x)∂xj∂uν(x)∂xs‖H2,μm-1(Ω)+‖∂a(x)∂xjuν(x)‖H2,μm-1(Ω)≤‖f(x)‖H2,μm(Ω)+(m-1)m⋅max⁡{2C1;2C2;C3}×(‖uν(x)‖H2,ν+β/2-1m+1(Ω)+‖uν(x)‖H2,ν+β/2-2m(Ω)+‖uν(x)‖H2,ν+β/2-3m-1(Ω)).
To strengthen this estimation, we will receive an inequality
(5.46)‖Fj(x)‖H2,μm-1(Ω)≤‖f(x)‖H2,μm(Ω)+(m-1)m⋅C9‖uν(x)‖H2,ν+β/2-1m+1(Ω).
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 uν(x) to space H2,ν+β/2-1m+1(Ω) (under the Theorem 4.2 and to integrated identity (5.13) which is fair for function uν(x) at ν and at ν-1). Thus, the estimation 5.72 takes the form
(5.47)‖uν(x)‖H2,ν+β/2-1m+1(Ω)≤C8(‖f(x)‖H2,μm-1(Ω)+‖φ(x)‖H2,μ+βm+1/2(∂Ω)).
From here by virtue of validity of inequalities
(5.48)‖f(x)‖H2,μm-1(Ω)≤max⁡∀x∈Ω¯ρ(x)‖f(x)‖H2,μm(Ω),‖φ(x)‖H2,μ+βm+1/2(∂Ω)≤max⁡∀x∈Ω¯ρ(x)‖φ(x)‖H2,μ+βm+3/2(∂Ω),
the estimation
(5.49)‖uν(x)‖H2,ν+β/2-1m+1(Ω)≤C8max⁡∀x∈Ω¯ρ(x)(‖f(x)‖H2,μm(Ω)+‖φ(x)‖H2,μ+βm+3/2(∂Ω))
follows. Therefore, function Fj(x) is limited in norm of space H2,μm-1(Ω), that is, Fj(x)∈H2,μm-1(Ω). Besides, for the right part of the differential equation (5.21) on the basic of inequalities (5.46) and (5.49) the estimation
(5.50)‖Fj(x)‖H2,μm-1(Ω)≤C14(‖f(x)‖H2,μm(Ω)+‖φ(x)‖H2,μ+βm+3/2(∂Ω))
will be true. Constant C14 is positive and does not depend from Fj(x), f(x), and φ(x).
Right partφj(x) of a boundary conditions (5.22) belongs to space H2,μ+βm+1/2(∂Ω). This fact at performance of a condition 5.42 directly follows from an inequality
(5.51)‖φj(x)‖H2,μ+βm+1/2(∂Ω)≤‖φ(x)‖H2,μ+βm+3/2(∂Ω).
Therefore, for a boundary value problem (5.21), (5.22) conditions at which function uνj(x) belongs to space H2,ν+β/2m+1(Ω) are satisfied all and the estimation
(5.52)‖uνj(x)‖H2,ν+β/2m+1(Ω)≤C8(‖Fj(x)‖H2,μm-1(Ω)+‖φj(x)‖H2,μ+βm+1/2(∂Ω))
is fair (according to the assumption of validity of the statement of the Theorem 5.1 at k=m-1).
Let us establish limitation seminorm|uν(x)|H2,ν+β/2m+2(Ω). For this purpose, we will estimate its square
(5.53)|uν(x)|2H2,ν+β/2m+2(Ω)=∑|λ|=m+2∫Ωρ2(ν+β/2)(x)|Dλuν(x)|2dx,
from above on inequality
(5.54)∑|λ|=m+2|Dλuν(x)|2≤∑|λ|=m+1|Dλ(∂uν(x)∂x1)|2+∑|λ|=m+1|Dλ(∂uν(x)∂x2)|2.
We will receive inequality
(5.55)|uν(x)|H2,ν+β/2m+2(Ω)2≤|uν1(x)|H2,ν+β/2m+1(Ω)2+|uν2(x)|H2,ν+β/2m+1(Ω)2.
From an inequality (5.55) and the function uνj(x) (j=1,2) proved above a belonging space H2,ν+β/2m+1(Ω) is followed with limitation seminorm, that is, uν(x) belongs to space H2,ν+β/2m+2(Ω).
Let us prove now an estimation5.73. For this purpose, we will consider function ρν+β/2(x)uν(x). This function belongs to space Hm+2(Ω), that follows from a belonging of function uν(x) to space H2,ν+β/2m+2(Ω) and statement “A” a lemma 1 of [6]. From statement “B” of the same lemma the inequality
(5.56)‖uν(x)‖H2,ν+β/2m+2(Ω)≤C15‖ρν+β/2(x)uν(x)‖Hm+2(Ω)
follows. Constant C15 is positive and does not depend from uν(x).
The norm of functionρν+β/2(x)uν(x) in Sobolev space Hm+2(Ω) can be determined the formula
(5.57)‖ρν+β/2(x)uν(x)‖Hm+2(Ω)=‖ρν+β/2(x)uν(x)‖L2(Ω)+|ρν+β/2uν(x)|Hm+2(Ω).
Let us estimate from above composed which enter into the right part of equality (5.57). For the first composed the estimation
(5.58)‖ρν+β/2(x)uν(x)‖L2(Ω)≤max⁡∀x∈Ω¯ρm+2(x)‖uν(x)‖L2,ν+β/2-m-2(Ω)
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 Hm+2(Ω), the formula Dm+2(g(x)·h(x))=∑i=0m+2Cm+2i(Dig(x))·(Dm+2-ih(x)), an algebraic inequality (∑q=1naq)2≤n·∑q=1naq2, and conditions which derivatives of weight function possess. The inequality
(5.59)|ρν+β/2(x)uν(x)|2Hm+2(Ω)≤(m+3)|uν(x)|2H2,ν+β/2m+2(Ω)+(m+3)(m+2)σ|uν(x)|H2,ν+β/2-1m+1(Ω)2+(m+3)(m+2)σ2×|uν(x)|H2,ν+β/2-2m(Ω)2+⋯+(m+3)σm+2‖uν(x)‖L2,ν+β/2-m-2(Ω)2
will be as a result received. From inequalities (5.56)–(5.59), the estimation
(5.60)‖uν(x)‖H2,ν+β/2m+2(Ω)2≤C162|uν(x)|H2,ν+β/2m+2(Ω)2+C172∥uν(x)∥H2,ν+β/2-1m+1(Ω)2
follows
(5.61)C162=2(m+3)C152,C172=2C152max⁡{max⁡∀x∈Ω¯ρm+2(x)+(m+3)σm+2;(m+3)(m+2)σm+1;⋯;(m+3)(m+2)σmax⁡∀x∈Ω¯ρm+2(x)}.
From it on an inequality a2+b2≤a+b which is fair for nonnegative values a and b, the inequality
(5.62)‖uν(x)‖H2,ν+β/2m+2(Ω)≤C16|uν(x)|H2,ν+β/2m+2(Ω)+C17‖uν(x)‖H2,ν+(β/2)-1m+1(Ω)
follows. From estimation (5.49)–(5.52), (5.62), the proved inequality 5.73 directly follows. Therefore, for k=m the statement of the Theorem 5.1 is proved.
So, the statement of the Theorem5.1 is proved for k=1 and also for k=m in the assumption at which it is true for k=m-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 k follows. The Theorem5.1 is proved.



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*Source: 243724-2011-07-05.xml*