On Differential Properties Rν-Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data

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 𝑅𝜈-generalized solution, that is, the fact that it belongs to the space 𝐻𝑘


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 u x does not belong to the Sobolev space W 1 2 H 1 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 an R ν -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 the R ν -generalized solution, that is, the fact that it belongs to the space H k 2 2,ν β/2 Ω .

The Basic Designations
Let R 2 denote the two-dimensional Euclidean space with x x 1 , x 2 .Let Ω be a bounded convex domain with boundary ∂Ω, and let Ω be the closure of Ω.We denote by ∂Ω o the set of the points of ∂Ω: ∂Ω i ∈ ∂Ω, ∂Ω o n i 1 ∂Ω i .Let us assume that boundary ∂Ω is a piecewise smooth and Let ρ x be a weight function that is infinitely differentiable and positive everywhere, except at the points of ∂Ω o , 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 where j 1, 2; i i 1 , i 2 , |i| i 1 i 2 are nonnegative integers; δ, σ, γ is real, δ > 1.We introduce the weighted Sobolev space H k 2,α 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
Seminorms in the this space look like where Denote by H k ∞,−α Ω, C the set of functions with the norm satisfying the inequality where

Statement of Problem: Definition of the R ν -Generalized Solution
Suppose that the differential equation is given in the domain Ω, with the boundary condition We suppose that a 12 x a 21 x and for some real where ξ 1 , ξ 2 are arbitrary real parameters, C i 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 a u, v are satisfied at some fixed integer k ≥ 1 and also 3.6 , 3.7 , 4.1 , and 4.4 hold.Then, the R νgeneralized solution u ν x of the Dirichlet problem with coordinated degeneration of the input data belongs to space H k 2 2, ν β/2 Ω and following estimate holds where C 8 is a positive constant independent of u ν 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 u ν x belongs to space H 2 2,ν β/2 Ω .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 k 1; at the second stage, we will prove it for k m the assumption that it is true at k m − 1.
Let us designate u νj x ∂u ν x /∂x j j 1, 2 .We will prove that functions u νj x belong to space H 1 2,ν β/2 Ω .For this purpose we will fix j and we will establish the upper bound a square of norm We consider the sum 5.9 We add to the right part of this equality nonnegative composed Then, we will receive an inequality

5.10
Similarly, the inequality turns out

5.11
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 From this estimation as We write down integrated identity a u ν , v f, v in the form of

5.13
In identity 5.13 , we will designate v x ∂v 1 x /∂x j j 1, 2 , where function v 1 x belongs to space o H 2 2,ν β/2 1 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
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

5.17
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

5.18
Difference Lu ν x − f x is equal to zero almost everywhere on Ω.The integrated identity will take the form holds.

ISRN Mathematical Analysis 9
Let us prove that function u * x belongs to space H 2 2,ν β/2 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 L 2,μ Ω .For this purpose, we will estimate composed in expression F j x .The first composed we will estimate in norm of space L 2,μ Ω

5.23
The second, the third, and the fourth composed at performance conditions 5.1 1 -5.3 1 we will estimate in norm of space

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 μ β/2, whence ν − β/2 μ.From this fact and four previous estimations, we will receive inequalities

5.25
Let us strengthen this inequality, and we will receive an estimation where C 9 max{2C 1 ; √ 2C 2 ; C 3 }.The first composed the right part of received estimation is limited under condition of 5.4 1 .Let us prove that second composed too is limited.For this purpose, we will notice that, by analogy with 6, Theorem 2 , the R ν -generalized solution of a boundary value problem 3.1 , 3.2 is unique same for various values of parameter ν.Therefore, function u ν 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 u ν x follows space H 2 2,ν β/2−1 Ω and also validity of an estimation 2,μ β ∂Ω .

5.27
From this estimation at performance of inequalities 2,μ β ∂Ω , 5.28 the estimation 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 Let us specify that the integrated identity 30 * is fair for all functions v * x from o H 1 2,ν β/2 Ω , therefore, it will be fair and for functions v 1 x .On the basic of this remark, identities 5.13 and 30 * , and also uniqueness of the R ν -generalized solution of a boundary value problem 5.21 , 5.22 it is received, that u νj x ≡ u * x , that is, function u νj x belongs to space H 2 2,ν β/2 Ω and for it the inequality 5.32 is fairly.Let us establish limitation seminorm |u ν x | H 3 2,ν β/2 Ω .For this purpose, we will estimate its square from above on inequality

5.34
We will receive inequality

5.35
From inequality 5.35 and function u νj x j 1, 2 proved above, a belonging space H 2 2,ν β/2 Ω is followed with limitation seminorm, therefore, u ν x ∈ H 3 2,ν β/2 Ω .Let us prove now an estimation 5.7 for k 1.For this purpose we will consider function ρ ν β/2 x u ν x : it belongs to space H 3 Ω .It follows from a belonging of function u ν x to space H 3 2,ν β/2 Ω and statements "A" of a lemma 1 of 6 .From statement "B" of the same lemma, the inequality follows.C 11 is a positive constant independent of u ν x .The norm for function ρ ν β/2 x u ν x in Sobolev space H 3 Ω can be entered it is equivalent see 12, page 380

5.37
Let us estimate from above composed the right part 5.37 .For the first composed truly an inequality

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 H 3 Ω , formula q and conditions which derivatives of weight function possess.As a result, we will receive an inequality

5.41
From estimations 5.29 -5.31 , 5.41 , and estimation 5.32 which is written down for u νj x ≡ u * x j 1, 2 , we will receive an inequality So, u ν x ∈ H 3 2,ν β/2 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 the R ν -generalized solution u ν x of the Dirichlet problem with coordinated degeneration of the input data belongs to space H m 1 2,ν β/2 Ω and the estimation we will prove that the R ν -generalized solution u ν x of the Dirichlet problem with coordinated degeneration of the input data belongs to space H m 2 2,ν β/2 Ω and the estimation 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 5.1 3 -5.6 3 conditions 5.1 2 -5.6 2 are carried out automatically, that is, u ν x ∈ H m 1 2,ν β/2 Ω and the estimation 5.1 2 -5.6 2 is fair.Again, we will take advantage of designations u νj x ∂u ν x /∂x j for j 1, 2 .To similarly how it has been made at a Stage 1, it is possible to show that function u νj x at fixed j is the R ν -generalized solution of a boundary value problem 5.21 , 5.22 and belongs to space H m 2,ν β/2 Ω .The last follows from an inequality
Let us prove that right part F j x of the differential equation 5.21 belongs to space H m−1 2,μ Ω .For this purpose, we will estimate composed the right part in equality F j x .The first composed we will estimate in norm of space

5.43
The second, the third, and the fourth composed we will estimate in norm of space H m−1 2,ν−β/2 Ω with use of conditions 5.1 3 -5.3 3 , formulas As a result, we will receive inequalities 2 l,s 1

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 μ β/2, hence ν − β/2 μ.From this fact and the four previous estimations, we will receive inequalities

5.45
To strengthen this estimation, we will receive an inequality

5.46
The first composed the right part of the received inequality is limited at performance of a condition 5.4 3 .Limitation of the second composed follows from a belonging of function u ν x to space H m 1 2,ν β/2−1 Ω under the Theorem 4.2 and to integrated identity 5.13 which is fair for function u ν x at ν and at ν − 1 .Thus, the estimation 5.7 2 takes the form 2, μ β ∂Ω .

5.47
From here by virtue of validity of inequalities follows.Therefore, function F j x is limited in norm of space H m−1 2,μ Ω , that is, F j x ∈ H m−1 2,μ 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 will be true.Constant C 14 is positive and does not depend from F j x , f x , and ϕ x .Right part ϕ j x of a boundary conditions 5.22 belongs to space H m 1/2 2,μ β ∂Ω .This fact at performance of a condition 5.4 2 directly follows from an inequality

5.51
Therefore, for a boundary value problem 5.21 , 5.22 conditions at which function u νj x belongs to space H m 1 2,ν β/2 Ω are satisfied all and the estimation .

5.54
We will receive inequality

5.55
From an inequality 5.55 and the function u νj x j 1, 2 proved above a belonging space H m 1 2,ν β/2 Ω is followed with limitation seminorm, that is, u ν x belongs to space H m 2 2,ν β/2 Ω .Let us prove now an estimation 5.7 3 .For this purpose, we will consider function ρ ν β/2 x u ν x .This function belongs to space H m 2 Ω , that follows from a belonging of function u ν x to space H m 2 2,ν β/2 Ω and statement "A" a lemma 1 of 6 .From statement "B" of the same lemma the inequality i 0 C i m 2 D i g x • D m 2−i h x , an algebraic inequality n q 1 a q 2 ≤ n • n q 1 a 2 q , and conditions which derivatives of weight function possess.The inequality 5.59 will be as a result received.From inequalities 5.56 -5.59 , the estimation  Therefore, for k m the statement of the Theorem 5.1 is proved.So, the statement of the Theorem 5.1 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 Theorem 5.1 is proved.
At the put-forward assumption of validity of the statement of the Theorem 5.1 for k m − 1, under conditions 3.6 , 3.7 , 4.1 , and 4.4 and under conditions

H m 2 Ω
u ν x H m 2 2,ν β/2 Ω ≤ C 15 ρ ν β/2 x u ν x H m 2 Ω5.56follows.Constant C 15 is positive and does not depend from u ν x .The norm of function ρ ν β/2 x u ν x in Sobolev space H m 2 Ω can be determined the formula ρ ν β/2 x u ν x from above composed which enter into the right part of equality 5.57 .For the first composed the estimationρ ν β/2 x u ν x L 2 Ω ≤ max ∀x∈Ω ρ m 2 x u ν x L 2,ν β/2−m−2 Ω5.58 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 H m 2 Ω , the formula D m 2 g x • h x m 2

. Existence, Uniqueness, and Coercive of the R ν -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

5. Differential Properties of an R ν -Generalized Solution of the Dirichlet Problem with Coordinated Degeneration of the Input Data Theorem 5.1. Let
a 12 x a 21 x and conditions ∂Ω .Let function u * x is the R ν -generalized solution of a boundary value problem 5.21 , 5.22 .Then, for any function