On the Existence and Uniqueness of R V-Generalized Solution for Dirichlet Problem with Singularity on All Boundary

and Applied Analysis 3 Denote by


Introduction
The singularity of solution for boundary value problems to two-dimensional closed domain can be due to the degeneration of the input data (coefficients and right-hand sides of equations and boundary conditions), availability of the reentrant corners, and change of the kind of the boundary conditions or by the internal properties of the solution.A boundary value problem is said to possess strong singularity if its solution () does not belong to Sobolev space  1  2 ( 1 ) or, in other words, the Dirichlet integral of the solution () diverges.In the case if the solution belongs to the space  1 2 ( 1 ) but does not belong to the space  2  2 ( 2 ), a boundary value problem is called the problem with a weak singularity.
Boundary value problems with strong singularity 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 strong singularity were constructed, based on separation of singular and regular components, mesh refinement near singular points, multiplicative extraction of singularities, and so forth, (see, e.g., [1][2][3][4][5][6]).
In [7], it was suggested to define the solution of boundary value problem for second-order elliptic equation with singularity on a finite set of points belonging to boundary of a two-dimensional domain as an  ] -generalized solution in the weighted Sobolev space.Such a new concept of solution led to the distinction of two classes of boundary value problems: problems with coordinated and uncoordinated degeneracy 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 [8,9]).
For boundary value problems for elliptic equations, Maxwell equations and Lamé system, we constructed the numerical methods with rate of convergence independent of the singularity based on the concept of an  ] -generalized solution (see, e.g., [10][11][12]).
In this paper, we consider the first boundary value problem for a second-order elliptic equation with strong singularity solution on all boundary of a two-dimensional domain.We distinguish two classes of the boundary value problems: problems with coordinated and uncoordinated degeneracy of input data.For this problem we define the solution as an  ] -generalized one in a weighted Sobolev space  1  2,]+/2 (Ω) and in a weighted set  1 2,]+/2 (Ω, ), respectively.We prove its existence and uniqueness in the corresponding weighted space and weighted set.It was established that, for all values of parameter ] for which the  ] -generalized solution exists, it is unique for all of these parameters.

Notation and Auxiliary Statements
We denote the two-dimensional Euclidean space by R 2 with  = ( 1 ,  2 ) and  =  1  2 .Let Ω ⊂ R 2 be a bounded domain with sufficiently smooth boundary Ω, and let Ω be the closure of Ω; that is, Ω = Ω ∪ Ω.We denote by Ω  the adjoining streak of the boundary Ω of width  > 0 and Ω  ⊂ Ω.
We introduce a weight function () that coincides in Ω  with the distance from point  to the boundary Ω and is equal to  for  ∈ Ω \ Ω  .
Proof.Taking into account condition (a), one can show that, for  * > , we have where  7 is a constant dependent of mes Ω  .Considering condition (b), we write the inequality for the function  as follows: From inequalities ( 5) and ( 6) we get the estimate (4) with  6 = ( 1 / 2 )√ 7 /2.

The Boundary Value Problem with Coordinated Degeneration of the Input Data on All Boundary of the Domain
In the domain Ω, we consider the differential equation with the boundary condition Definition 2. The boundary value problem (7) and ( 8) is called the Dirichlet problem with coordinated degeneration of the input data on all boundary of the domain or Problem A, if   () =   () (,  = 1, 2) and, for some real number , () >  12  −2 () almost everywhere on Ω and right-hand side of (7) satisfies where   ( = 8, . . ., 12) are positive constants independent of ;  1 and  2 are any real parameters;  is some nonnegative real number.
Denote by the bilinear and linear forms, respectively.
holds, where ] is arbitrary but fixed and satisfies the inequality For Problem A, we prove the main result.
Theorem 4. Let conditions (9)-( 12) and (15) hold and let where  13 is a positive constant not depending on  ] and .

Definition 3 . 1 2 1 2
A function  ] from the space ∘  ,]+/2 (Ω) is called an  ] -generalized solution of the Dirichlet problem with coordinated degeneration of the input data on all boundary of the domain or Problem A, if, for any V in ∘  ,]+/2 (Ω), the identity ] -generalized solution  ] of the Dirichlet problem with coordinated degeneration of the input data on all boundary of the domain exists and is unique in the space

The Boundary Value Problem with Uncoordinated Degeneration of the Input Data on All Boundary of the Domain
If there exists at least one ] for which there exists a unique  ] -generalized solution of the Problem A, then one can always define a half-open interval[] 1 , ] 2 ) such that, for each ] ∈ [] 1 , ] 2 ), there exists a unique  ] -generalized solution.Here, If the assumptions of Theorem 4 are valid, then, for all ] in the interval [] 1 , ] 2 ), the  ] -generalized solution of the Problem A is unique. 4.