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V. Rukavishnikov, E. Rukavishnikova, "On the Existence and Uniqueness of -Generalized Solution for Dirichlet Problem with Singularity on All Boundary", Abstract and Applied Analysis, vol. 2014, Article ID 568726, 6 pages, 2014. https://doi.org/10.1155/2014/568726
On the Existence and Uniqueness of -Generalized Solution for Dirichlet Problem with Singularity on All Boundary
The existence and uniqueness of the -generalized solution for the first boundary value problem and a second order elliptic equation with coordinated and uncoordinated degeneracy of input data and with strong singularity solution on all boundary of a two-dimensional domain are established.
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 () or, in other words, the Dirichlet integral of the solution diverges. In the case if the solution belongs to the space () but does not belong to the space (), 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., [1–6]).
In , 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., [10–12]).
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 and in a weighted set , 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.
2. Notation and Auxiliary Statements
We denote the two-dimensional Euclidean space by with and . Let be a bounded domain with sufficiently smooth boundary , and let be the closure of ; that is, . We denote by the adjoining streak of the boundary of width and .
We introduce a weight function that coincides in with the distance from point to the boundary and is equal to for .
Let and be the weighted spaces with norms: where , , and ; are integer nonnegative numbers, is some real nonnegative number, and is an integer nonnegative number. For we use the notation .
By for , we denote a set of functions satisfying the following conditions:(a) for , where , is positive constant independent of ,(b),and with the norm (2).
Let be the set of functions with the norm satisfying the inequality with a positive constant independent of . For , we have .
Lemma 1. For each function in the set and for any , the estimate holds, where , .
Proof. Taking into account condition (a), one can show that, for , we have where is a constant dependent of . Considering condition (b), we write the inequality for the function as follows: From inequalities (5) and (6) we get the estimate (4) with .
3. The Boundary Value Problem with Coordinated Degeneration of the Input Data on All Boundary of the Domain
In the domain , 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 and, for some real number , and right-hand side of (7) satisfies where () are positive constants independent of ; and are any real parameters; is some nonnegative real number.
Denote by the bilinear and linear forms, respectively.
Definition 3. A function from the space 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 in , the identity 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
Then, the -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 and the following estimate is valid: where is a positive constant not depending on and .
Proof. First, we show that the forms and are continuous on . In fact, by virtue of conditions (9), (12), and (15) and the Cauchy-Schwarz inequality, we have
Let us now prove the ellipticity of the bilinear form ; that is, We substitute by in (13), and by means of the Cauchy-Schwarz inequality, -inequality, and conditions (9), we estimate the absolute values of the second, third, and fourth terms of the form : Here, , , and are any positive numbers.
Using (10) and (11), we have Then, from (22)–(26), we obtain
Note that if condition (16) is satisfied, then there exists a positive constant such that Supposing that in (27), we get with constant
According to (19), (21), and (20), the bilinear form is continuous and -elliptical, and the linear form is continuous on ; then, the existence and uniqueness of an -generalized solution of Problem A follow from the Lax-Milgram theorem (see ).
Taking into account that we get estimate (17).
Corollary 5. 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 such that, for each , there exists a unique -generalized solution. Here, where is a given sufficiently small positive number.
Corollary follows from the proof of Theorem 4.
Theorem 6. If the assumptions of Theorem 4 are valid, then, for all in the interval , the -generalized solution of the Problem A is unique.
The proof of this statement is similar to that of Theorem 2 in .
4. The Boundary Value Problem with Uncoordinated Degeneration of the Input Data on All Boundary of the Domain
We consider the boundary value problem
Definition 7. The boundary value problem (33) and (34) is called the Dirichlet problem with uncoordinated degeneration of the input data on all boundary of the domain or Problem B, if, for some real number , and the right-hand side of the equation satisfies the condition for some nonnegative real number . Here, , , are positive constants not depending on ; and are arbitrary real parameters.
Definition 8. A function from the set is called an -generalized solution of the Problem B if the identity holds for all and for any given value of satisfying the inequality
Theorem 9. Let conditions (35)–(40) hold and Then, for any satisfying conditions (40) and (41), there always exists parameter such that -generalized solution of the Dirichlet problem with uncoordinated degeneration of the input data on all boundary of the domain exists and is unique in the set . In this case, the following estimate is valid: where is a positive constant independent of and .
Proof. First of all, we note that the bilinear and linear forms are continuous on the set and the inequalities
hold. The proofs of estimates (43) and (44) are established by analogy with (19) and (20), which we obtain by using conditions (35), (38), and (40) and Lemma 1.
Let us show that the bilinear form is -elliptical in . We have for any from . By means of condition (35), we estimate the absolute value of the second term on the right-hand side in (45): From (45) and the last inequality we get Supposing that and equal and in Lemma 1, respectively, we have Taking into account (36), (37), and (48), from estimate (47) we get Obviously, we can always choose and such that the constants , and the inequality are valid with constant . Therefore, bilinear form is -elliptical.
According to (43), (44), and (50), the bilinear form is continuous and -elliptical, and the linear form is continuous on ; then, the existence and uniqueness of an -generalized solution of Problem B follow from the Lax-Milgram theorem (see ).
Taking into account that we get estimate (42).
Theorem 10. If for some there is a set of values such that an -generalized solution of the Problem B exists in the set , then this solution is unique for all such .
The proof of this statement is similar to that of Theorem 2 in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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