ISRN Mathematical Analysis

Volume 2011 (2011), Article ID 243724, 18 pages

doi:10.5402/2011/243724

Research Article

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

Victor Anatolievich Rukavishnikov

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. Misiołek and C. Zhu

Copyright © 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 -generalized solution, that is, the fact that it belongs to the space

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 does not belong to the Sobolev space () or, in other words, the Dirichlet integral of the solution 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 -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], and - 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 -generalized solution, that is, the fact that it belongs to the space .

2. The Basic Designations

Let denote the two-dimensional Euclidean space with . Let be a bounded convex domain with boundary , and let be the closure of . We denote by the set of the points of : , . Let us assume that boundary is a piecewise smooth and.

Let be a weight function that is infinitely differentiable and positive everywhere, except at the points of , with coinciding in some neighborhood of each point with the distance to . Moreover, the derivatives of satisfy the inequality

where , are nonnegative integers; , , is real, .

We introduce the weighted Sobolev space which at fixed integer and real is updating of the set infinitely differentiated in function on norm

Seminorms in the this space look like

where is some integer, . As usual, .

Let us count function from space if there is such function from space that

Denote by the set of functions with the norm satisfying the inequality

where is a positive constant independent of .

As usual, .

3. Statement of Problem: Definition of the -Generalized Solution

Suppose that the differential equation

is given in the domain , with the boundary condition

We suppose that and for some real

where , are arbitrary real parameters, are positive constants independent of , and 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

Definition 3.1. A function in the space will be called -generalized solution of the Dirichlet problem with coordinated degeneration of the input data if almost everywhere in and for all in , the identity holds, where is arbitrary but fixed and satisfies the inequality

4. Existence, Uniqueness, and Coercive of the -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

holds. Then there exists a unique -generalized solution of the Dirichlet problem with coordinated degeneration of the input data in the space and the following estimate holds:

where is a positive constant independent of , , and .

Theorem 4.2. Assume that conditions (3.3)–(3.8), (4.1) are satisfied and also

holds. Then -generalized solution of the Dirichlet problem with coordinated degeneration of the input data belongs to space and coercivity inequality is valid

where is a positive constant independent of , , and .

For case , the proof of the Theorem 4.2 resulted in [6, Theorem 3].

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

Theorem 5.1. Let and conditions

are satisfied at some fixed integer and also (3.6), (3.7), (4.1), and (4.4) hold. Then, the -generalized solution of the Dirichlet problem with coordinated degeneration of the input data belongs to space and following estimate holds

where is a positive constant independent of , , and .

Proof. At performance of conditions (5.1)–(5.6) of the Theorem 5.1 conditions of the Theorem 4.2 are satisfied, therefore, function belongs to space .

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 ; at the second stage, we will prove it for the assumption that it is true at .

Stage 1. Let conditions (3.6), (3.7), (4.1), and (4.4) be satisfied and also conditions

holds.

Let us designate . We will prove that functions belong to space . For this purpose we will fix and we will establish the upper bound a square of norm

We consider the sum

We add to the right part of this equality nonnegative composed , , , . Then, we will receive an inequality

Similarly, the inequality turns out

Let’s take advantage of designations for and two previous inequalities. Then, from (5.8), we will receive an estimation

From this estimation (as ) limitation of function in norm of space follows, that is, .

We write down integrated identity in the form of

In identity (5.13), we will designate , where function belongs to space .

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 if ) and also the formula of differentiation of product of functions. We will receive equality

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

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

Difference is equal to zero almost everywhere on . The integrated identity will take the form

or , where

Suppose that the differential equation

is given in the domain , with the boundary condition

where .

Let function is the -generalized solution of a boundary value problem (5.21), (5.22). Then, for any function the integrated identity

holds.

Let us prove that function belongs to space . 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 . For this purpose, we will estimate composed in expression . The first composed we will estimate in norm of space

The second, the third, and the fourth composed at performance conditions – we will estimate in norm of space

The received estimations are fair for all values of parameter which satisfy condition (4.4). Therefore, they will be fair and for , whence . From this fact and four previous estimations, we will receive inequalities

Let us strengthen this inequality, and we will receive an estimation

where .

The first composed the right part of received estimation is limited under condition of . Let us prove that second composed too is limited. For this purpose, we will notice that, by analogy with [6, Theorem 2], the -generalized solution of a boundary value problem (3.1), (3.2) is unique (same) for various values of parameter . Therefore, function will satisfy integrated identity (5.13) at and at . From here and from the Theorem 4.2, the belonging of function follows space and also validity of an estimation

From this estimation at performance of inequalities

the estimation

follows. Therefore, function is limited in norm of space , that is, . 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

where is a positive constant independent of , and .

The right part of a boundary condition (5.22) belongs to space . Validity of this fact under condition follows from inequality

This inequality is obvious if to use definition of norm in space .

Therefore, at performance of conditions – and 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 belongs to space are satisfied all and the estimation

is fair. Besides, under the Theorem 4.1 by virtue of belonging it is received that function (the -generalized solution of a boundary value problem (5.21), (5.22)) exists and unique in space .

Let us specify that the integrated identity is fair for all functions from   , therefore, it will be fair and for functions . On the basic of this remark, identities (5.13) and , and also uniqueness of the -generalized solution of a boundary value problem (5.21), (5.22) it is received, that , that is, function belongs to space and for it the inequality (5.32) is fairly.

Let us establish limitation seminorm . For this purpose, we will estimate its square

from above on inequality

We will receive inequality

From inequality (5.35) and function () proved above, a belonging space is followed with limitation seminorm, therefore, .

Let us prove now an estimation (5.7) for . For this purpose we will consider function : it belongs to space . It follows from a belonging of function to space and statements “A” of a lemma 1 of [6]. From statement “B” of the same lemma, the inequality

follows. is a positive constant independent of .

The norm for function in Sobolev space can be entered it is equivalent (see [12, page 380])

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

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 , formula , algebraic inequality and conditions which derivatives of weight function possess. As a result, we will receive an inequality

From estimations (5.36)–(5.39), the inequality

follows. , .

From this estimation if to apply inequality which is true for nonnegative and , we will receive

From estimations (5.29)–(5.31), (5.41), and estimation (5.32) which is written down for (), we will receive an inequality

So, , the estimation (5.7) for is carried out. Therefore, the statement of the Theorem 5.1 is true for .

Stage 2. Let us assume that under conditions (3.6), (3.7), (4.1), and (4.4), and conditions

the -generalized solution of the Dirichlet problem with coordinated degeneration of the input data belongs to space and the estimation

is fair.

At the put-forward assumption of validity of the statement of the Theorem 5.1 for , under conditions (3.6), (3.7), (4.1), and (4.4) and under conditions

we will prove that the -generalized solution of the Dirichlet problem with coordinated degeneration of the input data belongs to space 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 – conditions – are carried out automatically, that is, and the estimation – is fair.

Again, we will take advantage of designations for . To similarly how it has been made at a Stage 1, it is possible to show that function at fixed is the -generalized solution of a boundary value problem (5.21), (5.22) and belongs to space . The last follows from an inequality

and belongings of function to space .

Let us prove that right part of the differential equation (5.21) belongs to space . For this purpose, we will estimate composed the right part in equality . The first composed we will estimate in norm of space

The second, the third, and the fourth composed we will estimate in norm of space with use of conditions –, formulas , and algebraic inequalities . As a result, we will receive inequalities

The received estimations are fair for all values of parameter which satisfy to condition (4.4). Therefore, they will be fair and for , hence . From this fact and the four previous estimations, we will receive inequalities

To strengthen this estimation, we will receive an inequality

The first composed the right part of the received inequality is limited at performance of a condition . Limitation of the second composed follows from a belonging of function to space (under the Theorem 4.2 and to integrated identity (5.13) which is fair for function at and at ). Thus, the estimation takes the form

From here by virtue of validity of inequalities

the estimation

follows. Therefore, function is limited in norm of space , that is, . Besides, for the right part of the differential equation (5.21) on the basic of inequalities (5.46) and (5.49) the estimation

will be true. Constant is positive and does not depend from , , and .

Right part of a boundary conditions (5.22) belongs to space . This fact at performance of a condition directly follows from an inequality

Therefore, for a boundary value problem (5.21), (5.22) conditions at which function belongs to space are satisfied all and the estimation

is fair (according to the assumption of validity of the statement of the Theorem 5.1 at ).

Let us establish limitation seminorm . For this purpose, we will estimate its square

from above on inequality

We will receive inequality

From an inequality (5.55) and the function () proved above a belonging space is followed with limitation seminorm, that is, belongs to space .

Let us prove now an estimation . For this purpose, we will consider function . This function belongs to space , that follows from a belonging of function to space and statement “A” a lemma 1 of [6]. From statement “B” of the same lemma the inequality

follows. Constant is positive and does not depend from .

The norm of function in Sobolev space can be determined the formula

Let us estimate from above composed which enter into the right part of equality (5.57). For the first composed the estimation

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 , the formula , an algebraic inequality , and conditions which derivatives of weight function possess. The inequality

will be as a result received. From inequalities (5.56)–(5.59), the estimation

follows

From it on an inequality which is fair for nonnegative values and , the inequality

follows. From estimation (5.49)–(5.52), (5.62), the proved inequality directly follows. Therefore, for the statement of the Theorem 5.1 is proved.

So, the statement of the Theorem 5.1 is proved for and also for in the assumption at which it is true for . 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 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.

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