GEOMETRIC SINGULARITIES OF THE POISSON’S EQUATION IN A NON-SMOOTH DOMAIN WITH APPLICATIONS OF WEIGHTED SOBOLEV SPACES

The solution fields of the elliptic boundary value problems may exhibit singularities near the corners, edges, crack tips, and so forth of the physical domain. This paper deals with the boundary singularities of weak solutions of boundary value problems governed by the Poisson equation in a two-dimensional non-smooth domain with singular points on the boundary. The presence of these points on the boundary, generally, generates local singularities in the solution. The applications of Fourier transform and weighted Sobolev spaces make it possible to describe the qualitative properties of the solution including its regularity. The general theory of V. A. Kondratiev is followed to obtain these results.


Introduction
Let D be a 2-dimensional bounded plane polygonal domain D ⊂ R 2 , (see Figure-1) whose boundary ∂D comprises the corner points (ω = π) and the points where the type of boundary conditions changes (ω = π).
Let N denote the set of these boundary points which consists of P 1 , ..., P N ⊂ ∂D. Note that a point P ∈ ∂D is said to be a corner point if there exists a neighborhood η(P ) of the point P such that D ∩ η(P ) is diffeomorphic to an angle κ intersected with the unit circle. Figure 1. A Polygonal domain.
Let P i denote the vertices of the polygon and the open edges Γ i connecting the vertices P i+1 and P i , 1 ≤ i ≤ N. Let P N +1 = P 1 , Γ N +1 = Γ 1 , Γ 0 = Γ N , and the interior angles are ω i = Γ i − Γ i−1 . Suppose that the boundary ∂D = Γ 0 ∪ Γ 1 and Γ 0 ∩ Γ 1 = ∅ with meas(Γ 0 ) > 0 (Lebesgue measure). Further, we assume that J 1 and J 2 be the disjoint subsets of 1, 2, ..., N , where we can set Γ 0 = i∈J1Γ i and Γ 1 = i∈J2Γ i , respectively, denote the union of the boundary parts, where the Dirichlet boundary conditions and the Neumann boundary conditions are given. As well, the combinations of the boundary points with different boundary conditions are considered. To characterize them, we denote the Dirichlet-Dirichlet boundary conditions by (DD), i.e., P i ∈ J 1 , the Neumann-Neumann boundary conditions by (NN), P i ∈ J 2 and the Dirichlet-Neumann (mixed) boundary conditions by (DN), P i ∈ J 1 J 2 . Note that N = J 1 ∪ J 2 ∪ J 1 J 2 .
To describe the problem mathematically, let us consider the mixed boundary value problem for the Poisson where n = (n 1 , n 2 ) is known as the unit outward normal vector to the boundary, f ∈ L 2 (D) and ∆ = ( ∂ 2 ∂x 2 + ∂ 2 ∂y 2 ) is the Laplacian operator. It is known from the theory of elliptic boundary value problems in domains with boundary irregularities, like corners, conic vertices, edges, and cracks, etc., the solution may exhibit singularities. Numerous interesting results about the regularity of the solution cannot be extended if one of the following situations appears: the domain has corners, edges or angular points on the boundary, the change of the boundary conditions at some points, the discontinuities of the solution and the singularities of the coefficients.
Principally, the theory for smooth domains cannot be applied directly to non-smooth domains having corners or edges on the boundary, and the points where the type of boundary conditions changes. The asymptotic expansion of the solution near the conical or angular points plays an important role to describe the regularity behavior of the solution accurately. Moreover, the information of the singularity functions in nonsmooth domains can help to improve the rate of convergence of the numerical methods for approximations, for instance, the finite element approximation, singular function method or the dual singular function method, and the graded mesh refinement [13,19,20]. Presently, there exists a wide-ranging theory for parabolic, hyperbolic, and elliptic boundary value problems having a smooth boundary. Generally, the results of this theory conclude that if the boundary of the domain, the boundary operators, the coefficients of the equations, and the right-hand sides are sufficiently smooth, then the solution of the considered problem is itself sufficiently smooth [12,13,18,29].
Generally, three types of singularities arise in elliptic type problems: the angular type singularities, the interface, and the infinity type singularities in unbounded solution domains. This paper deals with the angular type singularities and several approaches to find these singularities are discussed in [5,10,27,30]. In recent, [4] has comprehensively discussed the methods to find the singular behavior of the solution structure of the elliptic boundary value problems in a polygonal domain with convex and non-convex vertices. Further, it is noted from the general theory on H 2 -regularity for linear elliptic boundary value problems [14,15,25], the general solution u for two or three-dimensional domain D with corner or edge singularities and any right-hand side function f ∈ L 2 (D) can be broken down as a sum of a singular and a regular part where u R ∈ H 2 (D). The second part is the locally acting singular part that is a combination of explicit model singular solutions s m and the unknown coefficients c m . The special singular functions s m rely on the geometry of the model problem, the differential operator, and the characteristic boundary conditions. The unknown coefficients c m relating to singularity functions are some real numbers or unique scalar constants which are stated as the stress intensity factors. The rigorous formulas for their derivations are of constant interest and a challenging task [7,13,14,25]. The mathematical analysis like well-posedness and regularity results of such type of elliptic boundary value problems in non-smooth domains have attracted many mathematicians and scientists to examine the singular behavior of the solution structure near the singular points [9,15,16,21,23].
The main purpose of this paper is the derivation and the computation of the singular terms of the solutions of the generalized boundary eigenvalue problem for the Poisson equation in a bounded plane polygonal domain with singular points on its boundary. The theory developed by Kondratiev [22,23] and further extended by [28] for scalar problems is used in the context of weighted Sobolev spaces. Generally, the Sobolev spaces are not suitable to define the regularity results of the boundary value problems in non-smooth domains. So, [22,28] have introduced weighted Sobolev spaces with Kondratiev type weights for parabolic and elliptic problems in polygonal domains. In [25], where the method of special ansatzes and spherical coordinates are used to calculate the singular terms for the Dirichlet problem of the Poisson equation.
Analogous to [6], where the Mellin transform and the method of the special ansatzes is used to obtain the asymptotic singular representations of the solution of the biharmonic operator on a bounded domain with angular corners. The technique of Fourier transform is used here to obtain the generalized form of the boundary eigenvalue problem for the Poisson equation with the mixed boundary conditions. The achieved eigenvalues and eigensolutions generate singular terms. The information about the singular terms allows us to evaluate the optimal regularity of the corresponding weak solution of the considered boundary value problem.
The rest of this paper is organized as follows: Section 2 is dedicated to present the weak formulation of the problem and introduce some function spaces. In Section 3, determine a parametric boundary eigenvalue problem with a complex parameter ξ, the Poisson equation is considered in an infinite cone with various combinations of the boundary conditions. Furthermore, the distribution of the eigenvalues and the eigenfunctions are discussed. In Section 4, the regularity and expansion results for the corresponding problem with various conditions are investigated. Some concluding remarks are given in the last Section 5.

Analytical Preliminaries
Besides the strong formulation, let us consider the weak formulation of the mixed boundary value problem The Lax-Milgram theorem [12][13][14] deduces that the variational problem (2.1) has a unique solution. Hence, we have to analyze the smoothness of the weak solution v and see how it depends on the size of the angle ω i , i = 1, ..., N .

2.1.
Weighted Sobolev spaces. To analyze the regularity results of the weak solution of the corresponding boundary value problem in a non-smooth domain with singular points, firstly, we introduce some function spaces in line with [1,11,22,28].
Let N be the set of singular points on the boundary, i.e., N ⊂ ∂D. Denote where the supp v is bounded. We assume that D β v be the multi-index notation for higher-order derivatives and in cartesian coordinates is defined by Let α = α 1 , ..., α N be an N −tuple of real numbers which satisfying 0 < α i < 1 for 1 ≤ i ≤ N . Therefore, the weight function is characterized by where m is an any integer and r i (x) = dist (x, P i ). Let W m, p α (D) be the weighted Sobolev spaces and is

The boundary value problem in an infinite cone
In this section, we will see the occurrence of the singular terms near the singular points and the structure which they have. So, to analyze these results, the following steps are followed.
(1) We localize the model problem in the neighborhood of the corner point or a point where the boundary conditions changes (known as a singular point), and then the model problem is considered in an infinite cone.
(2) The model problem is transformed in the form of local polar coordinates (r, θ) and then the variable transformation r = e τ is used. Afterward, the complex Fourier transform respecting the variable τ is applied to attain a boundary value problem which depends on the complex parameter ξ. Moreover, the operatorV(ξ) is used to represent the generalized form of this parametric boundary eigenvalue problem.
is the neighborhood of the corner points or the points where the type of boundary conditions changes. Usually, these points are called singular points. To show that the weak solution v is regular, we have to investigate its behavior near the corner points P i , i = 1, 2, ..., N . Let us consider one corner point P N as an origin with an angle ω 0 , and an appropriate infinite differentiable cut-off function χ(|x|) = χ(r) is defined as which coincides with the original problem near the corner point P N . Then the system (1.1) become where F = χ f − 2∇χ · ∇v − v ∆χ and G(x) = 0 for r < and r > 2 . The behavior of u near the point P N illustrate the regularity of the solution v in the neighborhood of P N . If we suppose that the right-hand side The following boundary conditions are prescribed on the subsequent edges Γ S, 0 (θ = 0) and Γ S, ω0 (θ = ω 0 ) of the cone (see Figure-2). Just one condition is considered per edge to differentiate between the mixed boundary conditions.
To obtain the boundary eigenvalue value problem, some basic properties of the complex Fourier transform respecting variable τ in line with [16,23,28] are described as 5) and the inverse Fourier transform is It defines an isomorphic mapping, i.e., Moreover, it is noted that if h 1 < h 2 and the following properties are satisfied Now, by applying (3.5) to (3.4) with respect to τ , the parametric boundary value problem for the unknown functionû is obtained that depend on the complex parameter ξ and holds in the interval I = (0, ω 0 ).
Consequently, the transformed form of (3.4) is (3.11) LetV(ξ) represent the operator of (3.11) and it maps fromV(ξ) : that the operatorV(ξ) can be defined for every boundary point in the sense of [2,3]. So, the operator V(ξ)(ξ, θ) = 0 is used to describe a generalized eigenvalue problem and the solvability of these type of problems is discussed in [24]. The operatorV(ξ) realizes an isomorphism for all ξ ∈ C apart from some isolated points (known as the eigenvalues ofV(ξ)). So, the resolvent R(ξ) = V (ξ) −1 is an operator-valued, meromorphic function of ξ has poles of finite multiplicity.
To compute the eigenvalues ξ µ (generally referred for multiple eigenvalues) and the corresponding eigenfunctions, we proceed as.
with the constant c is independent of ξ.

The Regularity results
In this section, the regularity results and the expansion of the solution u or v of the boundary value problem (3.2) or (1.1) are defined.
To analyze the regularity results of the boundary value problem (3.11), the combinations of the boundary points with different boundary conditions are considered. First of all, it is to be determined that the righthand sides functions in (3.4) are Fourier transform in the sense of (3.7). We know from (3.2) that F ∈ L 2 (S), and further note that for all α ≥ 0, F ∈ W 0, 2 α (S). Since, F ∈ W 0, 2 α (S), we have where h = α − 1 for all α ≥ 0 and it is meaningful according to (3.7). Consequently, the Fourier transform ofF (τ, θ) is meaningful in the half plane h = Im ξ ≥ −1 for almost all θ ∈ (0, ω 0 ).
The following regularity results of the boundary value problem (3.2) for various combinations of the boundary conditions are achieved as a direct consequence of Theorem 3.2 and the contemplations in Section 3.
If no eigenvalues ofV(ξ) are lie on the line h = Im ξ = α − 1, α ≥ 0, then the inverse Fourier transform which can be read as follows in formula (3.6) exists andũ h (τ, θ) = u h (x) is the unique solution of (4.12) from W 2, 2 α (S). It follows from the theory of Kondratiev in [22,23], a regularity result yields that u ∈ W 2, 2 γ+1 (S) where γ is a small positive real number.
To derive an expansion of the solution u(x) in S, where v ∈ W 1,2 (D) is the unique weak solution of the boundary value problem (1.1). The main question is the inverse Fourier transformation of the right-hand sides of (4.12) which can be read as follows (4.14) The integral (4.14) can be calculated in the same way by considering the Cauchy theorem and the approach used for calculating the regularity results of Dirichlet boundary conditions.
Hence, we conclude that for ω 0 > π, the following expansion of the solution of the boundary value (3.2) is obtained where w(x) ∈ W 2, 2 0 (S) and u(x) ∈ W 1, 2 0 (S) is the solution of the boundary value (3.2). Now, substituting r = e τ , we get
Again we have, if no eigenvalues ofV(ξ) lie on the line h = Im ξ = α − 1, α ≥ 0, then the inverse Fourier transform which can be read as follows exists and u h (x) is the uniquely determined solution from W 2, 2 α (S) of (3.2) for mixed conditions (ND). From [22,23], a regularity result yields that u ∈ W 2, 2 γ+1 (S) where γ is a sufficiently small positive real number.
Finally, (4.24) completely describe the regularity of the solution v of the boundary value problem (1.1).

Conclusion
It is well-known from the theory of elliptic boundary value problems in domains with boundary irregularities, like corners, conic vertices, edges, and cracks, etc., the solution may exhibit singularities. Generally, the flows over corners usually change their behaviors and properties as a result of a rapid geometrical change in the shape. In this article, we have studied the boundary singularities and regularity of the weak solution of the mixed boundary value problem for the Poisson equation in a non-smooth domain with singular points on the boundary. The singular structure of the solution of the considered problem near the corner points is investigated through the Fourier transform and the suitable weighted Sobolev spaces that best characterize the singular behavior of the solution are presented. It is observed for Dirichlet and Neumann boundary conditions that if D has reentrant corners (ω i > π : i = 1, 2, ...N ), then the weak solution v ∈ W 1,2 0 (D) of the considered problem has the form (4.11) and (4.16). If the domain D is a convex polygonal domain, then the solution v ∈ W 2,2 (D). For the mixed boundary conditions, the general solution is presented in the form of (4.22). Moreover, it is shown that the solution of the given problem can be decomposed into the singular and regular parts near the corner points for the values of ω i ∈ ( π 2 , 3π 2 ) and ω i > 3π 2 and does not belong locally to space H 2 . Finally, (4.24) completely describe the regularity result of the original boundary value problem in a domain D with singular points on the boundary.
The results to be achieved here can be further extended to three-dimensional domains, for instance, polyhedral domain, etc. with straight edges to analyze the edge singularities and the regularity expansion of the solutions. Additionally, the technique to be presented here can be modified for investigating and treating numerous linear boundary value problems in two-dimensional domains with corners, such as Lame's equations, Stokes equations and so forth.

Conflict of Interest:
The author declares that no conflict of interest regarding the publication of this paper.