International Journal of Analysis and Applications Volume 18, Number 4 (2020), 672-688 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-672 GEOMETRIC SINGULARITIES OF THE POISSON’S EQUATION IN A NON-SMOOTH DOMAIN WITH APPLICATIONS OF WEIGHTED SOBOLEV SPACES YASIR NADEEM ANJAM1,2,∗ 1School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R.China 2Department of Applied Sciences, National Textile University, Faisalabad 37610, Pakistan ∗Corresponding author: ynanjam@ntu.edu.pk Abstract. 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 singular- ities 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. 1. Introduction Let D be a 2-dimensional bounded plane polygonal domain D ⊂ R2, (see Figure-1) whose boundary ∂D comprises the corner points (ω 6= 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 { P1, ...,PN } ⊂ ∂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. Received April 24th, 2020; accepted May 14th, 2020; published June 3rd, 2020. 2010 Mathematics Subject Classification. primary 35Q99; secondary 35J05, 35B65, 35J25. Key words and phrases. Poisson equation; mixed boundary conditions; non-smooth domain; regularity; weighted Sobolev spaces. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 672 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-672 Int. J. Anal. Appl. 18 (4) (2020) 673 N P 1 P 2 P 3 P 1  N  2  3  i  1  2  3  Figure 1. A Polygonal domain. Let Pi denote the vertices of the polygon and the open edges Γi connecting the vertices Pi+1 and Pi, 1 ≤ i ≤ N. Let PN+1 = P1, ΓN+1 = Γ1, Γ0 = ΓN , and the interior angles are ωi = Γi − Γi−1. Suppose that the boundary ∂D = Γ0 ∪ Γ1 and Γ0 ∩ Γ1 = ∅ with meas(Γ0) > 0 (Lebesgue measure). Further, we assume that J1 and J2 be the disjoint subsets of { 1, 2, ...,N } , where we can set Γ0 = ⋃ i∈J1 Γ̄i and Γ1 = ⋃ i∈J2 Γ̄i, 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., Pi ∈ J1, the Neumann-Neumann boundary conditions by (NN), Pi ∈ J2 and the Dirichlet-Neumann (mixed) boundary conditions by (DN), Pi ∈J1J2. Note that N = J1 ∪J2 ∪J1 J2. To describe the problem mathematically, let us consider the mixed boundary value problem for the Poisson equation   −∆v = f in D, v = 0 on Γ0, ∂v ∂n = 0 on Γ1, (1.1) where n = (n1,n2) is known as the unit outward normal vector to the boundary, f ∈ L2(D) and ∆ = ( ∂ 2 ∂x2 + ∂ 2 ∂y2 ) 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 interest- ing 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 Int. J. Anal. Appl. 18 (4) (2020) 674 regularity behavior of the solution accurately. Moreover, the information of the singularity functions in non- smooth 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 H2-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 ∈ L2(D) can be broken down as a sum of a singular and a regular part u = N∑ m=1 cmsm + uR, (1.2) where uR ∈ H2(D). The second part is the locally acting singular part that is a combination of explicit model singular solutions sm and the unknown coefficients cm. The special singular functions sm rely on the geometry of the model problem, the differential operator, and the characteristic boundary conditions. The unknown coefficients cm 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. Int. J. Anal. Appl. 18 (4) (2020) 675 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 eigen- functions 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. 2. Analytical Preliminaries Besides the strong formulation, let us consider the weak formulation of the mixed boundary value problem (1.1) which reads: Find v ∈ U(D) = {v ∈ H1(D) : v = 0 on Γ0} such that a(v,u) = f(v) ∀u ∈ U(D), (2.1) where a(v,u) = ∫ D ∇v ·∇udx and f(v) = ∫ D f ·udx. The Lax-Milgram theorem [12–14] 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 ωi, 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 C∞N = { v ∈ C∞(D), supp v ∩N = ∅ } , where the supp v is bounded. We assume that Dβv be the multi-index notation for higher-order derivatives and in cartesian coordinates is defined by Dβv = ∂|β|v ∂x β1 1 ∂x β2 2 , β = (β1, β2), |β| = β1 + β2. Int. J. Anal. Appl. 18 (4) (2020) 676 Let α = ( α1, ...,αN ) be an N−tuple of real numbers which satisfying 0 < αi < 1 for 1 ≤ i ≤ N. Therefore, the weight function is characterized by Φα+m(x) = N∏ i=1 ( ri(x) )αi+m , where m is an any integer and ( ri(x) ) = dist (x,Pi). Let Wm,pα (D) be the weighted Sobolev spaces and is the closure of C∞N (D) equipped with the norm ‖v‖Wm,pα (D) = ( ∑ |β|≤m ∫ D |x|p(α−m+|β|) ∣∣Dβv∣∣pdx)1p . (2.2) Let Q = { (τ,θ) : −∞ < τ < ∞, 0 < θ < ω0 } denote the infinite strip with positive width ω0. For any real h > 0, and for an integer m ≥ 0, the spaces are defined as Wmh (Q) = { u ∈ L2(Q) : ∑ |β|≤m ∫ Q e2hτ ∣∣Dβu∣∣2 dτdθ < ∞}, where ‖u‖Wm h (Q) = ( ∑ |β|≤m ∫ Q e2hτ ∣∣Dβu∣∣2 dτ dθ)12 . 3. 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 = eτ 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 operator V̂(ξ) is used to represent the generalized form of this parametric boundary eigenvalue problem. (3) The eigenvalues and the generalized eigensolutions of this parametric boundary eigenvalue problem with various kinds of boundary conditions are obtained. They exhibit the asymptotic development of the solution of the model problem near the singular points. Finally, the regularity results can be followed by the general theory of ellipticity. Int. J. Anal. Appl. 18 (4) (2020) 677 3.1. Localization of the model problem. The regularity analysis of the solutions of mixed boundary value problems in a bounded plane polygonal domain is a local problem. If we presume that D is a polygonal domain, then the regularity principles work well in the interior of the domain and also on ∂D\ ⋃N i=1 η(Pi), where η(Pi) 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 Pi, i = 1, 2, ...,N. Let us consider one corner point PN as an origin with an angle ω0, and an appropriate infinite differentiable cut-off function χ(|x|) = χ(r) is defined as χ(r) =   1 for 0 ≤ r ≤ �, 0 for r ≥ 2�, (3.1) and it depends on the distance from the point PN . The number � is so small that PN is the only corner point of the domain D that lies inside the circle {x : |x| ≤ 2�}. Afterward, multiplying the smooth cut-off function χ on both sides of (1.1), then substituting u = χv in (1.1). The derivatives are considered in the distribution sense. Thus, the boundary value problem is transformed into an infinite cone S = { (r, θ) : 0 < r < ∞, 0 < θ < ω0 } , which coincides with the original problem near the corner point PN . Then the system (1.1) become S NP 0 ,S w ,0S 0  Figure 2. Infinite cone S with opening angle ω0.   −∆u = F in S, u = 0 on ΓS, 0, ΓS,ω0 if ΓS, 0, ΓS,ω0 ⊂ Γ0, ∂u ∂n = G on ΓS, 0, ΓS,ω0 if ΓS, 0, ΓS,ω0 ⊂ Γ1, (3.2) where F = χf − 2∇χ ·∇v−v ∆χ and G(x) = 0 for r < � and r > 2�. The behavior of u near the point PN illustrate the regularity of the solution v in the neighborhood of PN . If we suppose that the right-hand side in (1.1) is f ∈ L2(D), then F ∈ L2(S). The following boundary conditions are prescribed on the subsequent Int. J. Anal. Appl. 18 (4) (2020) 678 edges ΓS, 0 (θ = 0) and ΓS,ω0 (θ = ω0) of the cone (see Figure-2). Just one condition is considered per edge to differentiate between the mixed boundary conditions. To analyze the regularity results of the boundary value problem (3.2), we rewrite the operators in the structure of polar coordinates (r,θ). Hence, the transformed form is − (∂2ǔ ∂r2 + 1 r ∂ǔ ∂r + 1 r2 ∂2ŭ ∂θ2 ) =F̌(r,θ) in Ŝ, ǔ(r,θ) ∣∣ θ=0,ω0 =0, ∂ǔ ∂θ (r,θ) ∣∣ θ=0,ω0 =Ǧ(r,θ) ∣∣ θ=0,ω0 , (3.3) where Ŝ is the infinite half-strip in the (r,θ)-plane and ǔ(r,θ) = u(x,y), F̌(r,θ) = F(x,y) and Ǧ(r,θ) = G(x,y). Now, a variable τ with the relation r = eτ is introduced, then (3.3) is transformed to the infinite strip with the width ω0 as − (∂2ũ ∂τ2 + ∂2ũ ∂θ2 ) =F̃(τ,θ) in S̄, ũ(τ,θ) ∣∣ θ=0,ω0 =0, ∂ũ ∂θ (τ,θ) ∣∣ θ=0,ω0 =G̃(τ,θ) ∣∣ θ=0,ω0 . (3.4) Here, S̄ = { (τ, θ) : −∞ < τ < ∞, 0 < θ < ω0 } and ũ = ǔ(eτ,θ), F̃ = e2τ F̌(eτ,θ) and G̃ = eτ Ǧ(eτ,θ). 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 F[u](ξ) = û(ξ) = (2π)− 1 2 ∫ ∞ −∞ e−iξτ u(τ)dτ, ξ ∈ C, (3.5) and the inverse Fourier transform is F−1[u](ξ) = u(τ) = (2π)− 1 2 ∫ ∞+ih −∞+ih eiξτû(ξ) dξ. (3.6) It defines an isomorphic mapping, i.e., F[u](ξ) = { u(τ) : ∫ ∞ −∞ e2hτ|u(τ)|2dτ < ∞ } → L2(R + ih), (3.7) for ξ = s + ih, where h = constant, R + ih = { ξ ∈ C : Im ξ = h } . Therefore, the subsequent Parseval identity holds ∫ ∞ −∞ e2hτ |u(τ)|2 dτ = ∫ ∞+ih −∞+ih |û(ξ)|2 dξ. (3.8) We have F ( dm dτm u(τ) ) (ξ) = (iξ)mF ( u(τ) ) (ξ). (3.9) Int. J. Anal. Appl. 18 (4) (2020) 679 Moreover, it is noted that if h1 < h2 and the following properties are satisfied∫ +∞ −∞ e2h1τ |u(τ)|2 dτ < ∞, ∫ +∞ −∞ e2h2τ |u(τ)|2 dτ < ∞, (3.10) then û(ξ) is holomorphic in the strip h1 < Im ξ < h2. Now, by applying (3.5) to (3.4) with respect to τ, the parametric boundary value problem for the unknown function û is obtained that depend on the complex parameter ξ and holds in the interval I = (0,ω0). Consequently, the transformed form of (3.4) is ξ2û− ∂2û ∂θ2 =F̂(ξ,θ), û(ξ,θ) ∣∣ θ=0,ω0 =0, ∂û ∂θ (ξ,θ) ∣∣ θ=0,ω0 =Ĝ(ξ,θ) ∣∣ θ=0,ω0 . (3.11) Let V̂(ξ) represent the operator of (3.11) and it maps from V̂(ξ) : W 2, 2(0,ω0) into L2(0,ω0) ×C×C. Note that the operator V̂(ξ) can be defined for every boundary point in the sense of [2, 3]. So, the operator V̂(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 operator V̂(ξ) realizes an isomorphism for all ξ ∈ C apart from some isolated points (known as the eigenvalues of V̂(ξ)). So, the resolvent R(ξ) = [ V̂(ξ) ]−1 is an operator-valued, meromorphic function of ξ has poles of finite multiplicity. To compute the eigenvalues ξµ (generally referred for multiple eigenvalues) and the corresponding eigen- functions, we proceed as. Definition 3.1. A complex number ξ = ξ0 is known as the eigenvalue of V̂(ξ) if there exists a nontrivial solution which is holomorphic at ξ0, i.e., û(.,ξ0) 6= 0, and V̂(ξ0) û(θ,ξ0) = 0, where û(θ,ξ0) is an eigenfunc- tion of V̂(ξ0) corresponding to the eigenvalue ξ0. The set of fields { û0(θ,ξ0), û0,1(θ,ξ0), ..., û0,s(θ,ξ0) } with û0,0 = û0 is said to be a Jordan chain corresponding to the eigenvalue ξ0, if the equation σ∑ q=0 1 q! ( ∂ ∂ξ )q V̂(ξ) û0,m−q(θ,ξ)∣∣∣ ξ=ξ0 = 0 for m = 1, 2, ...,s, is satisfied. The number s + 1 is called the length of the Jordan chain. Remark 3.1. It is noted from [22–24] that if the complex number ξ is not an eigenvalue of the operator V̂(ξ), then V̂(ξ) is an isomorphism among the spaces V̂(ξ) : W 2, 2(0,ω0) and L2(0,ω0) ×C×C. Theorem 3.1. Let lh = {ξ ∈ C : Im ξ = h}. If no eigenvalues of V̂(ξ) lies on the line lh, then the system (3.11) admits a unique solution û ∈ W 2,2(0,ω0) provided (F̂, 0,Ĝ) ∈ L2(0,ω0) × C × C, and it holds for all Int. J. Anal. Appl. 18 (4) (2020) 680 ξ ∈ lh: ‖û‖2W2,2(0,ω0) ≤ c { ‖F̂‖2L2(0,ω0) + |ξ||Ĝ| 2 } , (3.12) with the constant c is independent of ξ. Proof. A similar theorem is proved in ( [17], Theorem 4.9). So, we omit its proof. � Taking note from the above-mentioned results, and [22,23,28], we can derive a fundamental regularity and expansion theorem, based on the Fourier transform, for the mixed boundary value problem for the Poisson equation in a two-dimensional bounded domain with singular points on the boundary. By considering the substitution Re α = −Im ξ − 2 for α ∈ C, it improves theorems ( [24], Theorem 8.2.1 and Theorem 8.2.2.) which are based on the Mellin transform technique and used for the solvability of the elliptic systems. Theorem 3.2. (Regularity and expansion theorem). Let α1 and α2 be real numbers and satisfying α1 − 1 < α2 < α1. Let v ∈ Wm, 2α1 (D) be a solution of the mixed boundary value problem (1.1) and f ∈ Wm1−2,pα2 (D) ∩W m−2, 2 α1 (D), where 1 ≤ p < ∞, m1 ≥ m ≥ 2 and α1 ≥ α2 ≥ 0. Subsequently, the following implications holds: (1) If the strip α2 + 2 p −m1 ≤ Im ξ ≤ α1 + 1 −m, is free of eigenvalues of the operator V̂(ξ), then the solution v ∈Wm1,pα2 (D) and holds the following estimate ‖u‖Wm1,pα2 (D) ≤ c(D)‖f‖Wm1−2,pα2 (D) . (2) Let ξ1,ξ2, ...,ξM be the eigenvalues of the operator V̂(ξ) and suppose that no eigenvalue lie on the lines Im ξ = α2 + 2 p −m1 and Im ξ = α1 + 1 −m. If the eigenvalues ξ1,ξ2, ...,ξM are situated in the strip α2 + 2 p −m1 < Im ξ < α1 + 1 −m, then the solution v admits the following expansion in the neighborhood Pδ of the corner point P , i.e., v = χ(r) [ M∑ µ=1 Iµ∑ σ=1 κµσ−1∑ κ=0 cµ,σ,κ Φµ,σ,κ(r, θ) ] + vr(r, θ), (3.13) where vr(r, θ) ∈ Wm1,pα2 (Pδ). Here, we set M be the number of eigenvalues of the operator V̂(ξ) in the strip, the constants cµ,σ,κ depends on the data and the singular functions, Iµ = dim Ker V̂(ξµ) represents the geometrical multiplicity of ξµ, κµσ is the length of the Jordan chains of V̂(ξµ), and the corresponding singular function is described as Φµ,σ,κ(r, θ) = r iξµ κ∑ j=0 (i log r)j j! ψσ,κ−jµ (θ), (3.14) where ψσ,κ−jµ (θ) is a canonical system of Jordan chains of V̂(ξ) respecting ξµ. It is noted from (3.13) and (3.14) that the eigenvalues ξµ = 0 does not yield singularities in the development of the solution in the neighborhood Pδ. Int. J. Anal. Appl. 18 (4) (2020) 681 It is recognized for elliptic boundary value problems that the eigenvalues of the operator V̂(ξ) which lies in the strip have a significant role in the regularity results. The assertions 1 and 2 of Theorem 3.2 represent the regularity and the expansion of the solution of the system (1.1) near the singular points. 3.2. The calculation of the eigenvalues. In this section, the eigenvalues and eigenfunctions of the bound- ary value problem (3.11) with relationships of various boundary conditions are evaluated. Generally, no unique solution exists for different boundary conditions, and the multiple solutions are marked with an in- dex µ or l. The computed eigenvalues ξl and corresponding eigenfunctions Φl(θ) of the considered problem with various conditions are defined as follows. Dirichlet boundary conditions (DD) For Dirichlet boundary conditions, the eigenvalues of the operator V̂(ξ) are ξl = i lπω0 , l = ±1,±2, ..., and the corresponding eigenfunctions are Φl(θ) = sin lπ ω0 θ. Neumann boundary conditions (NN) For Neumann boundary conditions, the eigenvalues of the operator V̂(ξ) are ξl = i lπω0 , l = 0,±1,±2, ..., and the corresponding eigenfunctions are Φl(θ) = cos lπ ω0 θ. Mixed boundary conditions (ND) Similar to the latter cases of the boundary conditions, the eigenvalues of the corresponding operator V̂(ξ) are ξl = i(l + 1 2 ) π ω0 , l = 0,±1,±2, ..., and the eigenfunctions are Φl(θ) = cos(l + 12 ) π ω0 θ. Remark 3.2. It is noted that if the versed boundary conditions are used which means that the Dirichlet condition is at θ = 0 and the Neumann condition is at θ = ω0, then the same eigenvalues of V̂(ξ) like the mixed boundary conditions (ND) are obtained but the corresponding eigenfunctions are Φl(θ) = sin(l+ 1 2 ) π ω0 θ. 4. 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 right- hand sides functions in (3.4) are Fourier transform in the sense of (3.7). We know from (3.2) that F ∈ L2(S), and further note that for all α ≥ 0, F ∈W0, 2α (S). Since, F ∈W0, 2α (S), we have∫ S |F(x)|2 |x|2 α dx = ∫ S̄ e2(τα+τ)|F̃(τ, θ)|2dτ dθ < ∞, (4.1) where h = α− 1 for all α ≥ 0 and it is meaningful according to (3.7). Consequently, the Fourier transform of F̃(τ,θ) is meaningful in the half plane h = Im ξ ≥−1 for almost all θ ∈ (0, ω0). The following regularity results of the boundary value problem (3.2) for various combinations of the bound- ary conditions are achieved as a direct consequence of Theorem 3.2 and the contemplations in Section 3. Int. J. Anal. Appl. 18 (4) (2020) 682 Dirichlet boundary conditions (DD) Let V̂(ξ) denote the operator of the problem (3.11) for the Dirichlet-Dirichlet boundary conditions (DD) and V̂(ξ) : W 2, 2(0,ω0) → L2(0,ω0) × C × C. If ξ is no eigenvalue of V̂(ξ), then for any F̂ ∈ L2(0,ω0) a unique weak solution û of (3.11) exists. We write û = V̂−1(ξ)[F̂, 0, 0], (4.2) where V̂−1(ξ) represent the inverse (or resolvent) operator and V̂−1(ξ) : L2(0,ω0) × C × C → W 2, 2(0,ω0). Moreover, the inverse Fourier transform of û produces the solution ũ(τ,θ) = u(x,y) of (3.2) and the subse- quent regularity result holds. If no eigenvalues of V̂(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 and ũh(τ, θ) = uh(x) ∈ W2, 2α (S). Further, uh(x) be the unique solution of (3.2) from W2, 2α (S). It follows from the theory of Kondratiev in [22, 23], a regularity result yields that u ∈W2, 21 (S). Therefore, we have u(x) = u0(x). Let us derive an expansion of the solution u(x) in S, the main question is the inverse Fourier transformation of the right-hand sides of (3.11) which can be read as follows ũh(τ, θ) = (2π) −1 2 ∫ ∞+ih −∞+ih eiξτ V̂−1(ξ)[F̂, 0, 0] dξ. (4.3) Using the Cauchy theorem, yields ũh(τ, θ) = (2π) −1 2 lim n→∞ {∫ −n+iδ −n+ih eiξτ V̂−1(ξ)[F̂, 0, 0] dξ + ∫ n+iδ −n+iδ eiξτ V̂−1(ξ)[F̂, 0, 0] dξ + ∫ n+ih n+iδ eiξτ V̂−1(ξ)[F̂, 0, 0] dξ } + 1 √ 2π 2πi ∑ −1 π, then ξ−1 = −i πω0 lies in this strip. Further from [26], the operator V̂−1(ξ) has the following expansion in the neighborhood of ξ−1, i.e., V̂−1(ξ) = q1 (ξ − ξ−1) + Γ(ξ), (4.5) Int. J. Anal. Appl. 18 (4) (2020) 683 where q1 and Γ(ξ) map L 2(0,ω0) × C × C into W 2, 2(0,ω0). The operator q1 behaves as the space of eigenfunctions of V̂(ξ) corresponding to ξ−1 and Γ(ξ) is holomorphic. Moreover, (3.10) and (4.1) imply that F̂(ξ,θ) is holomorphic respecting ξ in the strip −1 < Im ξ < 0. Hence, we can write [F̂, 0, 0] = ∞∑ m=0 bm(θ) (ξ − ξ−1)m, (4.6) in a neighborhood of ξ−1, where the coefficients bm(θ) are elements of L 2(0,ω0) ×C×C. Finally, we have eiξτ = eiξ−1 [ 1 + i(ξ − ξ−1)τ + ... + [i(ξ − ξ−1)τ]m m! + ... ] . (4.7) From (4.5)-(4.7), it follows that eiξτ V̂−1(ξ) [ F̂, 0, 0 ] = eiξ−1τ [ 1 + ... ] ∞∑ m=0 [ q1bm(θ) (ξ − ξ−1)m (ξ − ξ−1) + Γ(ξ)bm(θ) (ξ − ξ−1)m ] . (4.8) We conclude that Res [ eiξτ V̂−1(ξ) [ F̂, 0, 0 ]]∣∣∣ ξ=ξ−1 = eiξ−1τq1b0(θ), = e πτ ω0 C1 sin π ω0 θ, (4.9) where C1 is a complex constant and the formulas for its precise computation can be found in [7, 8, 20]. For ω0 > π, (4.4) yields u(x) = ũ0(τ, θ) = e πτ ω0 C1 sin π ω0 θ + w(x), (4.10) where w(x) ∈ W2, 20 (S) and u(x) ∈ W 1, 2 0 (S) is the solution of the boundary value (3.2). Now, substituting r = eτ , we get u(x) = ũ0(τ, θ) = C1 r π ω0 sin π ω0 θ + w(x). (4.11) If ω0 ≤ π, then u(x) = w(x) ∈W 2, 2 0 (S). Neumann boundary conditions (NN) Let Pi be a boundary point at which the Neumann-Neumann (NN) conditions appear. Using the same approach which is used for the Dirichlet-Dirichlet conditions, the following Fourier transformed form of (3.2) is obtained as ξ2û− ∂2û ∂θ2 =F̂(ξ,θ), ∂û ∂θ (ξ, 0) =Ĝ(ξ, 0), ∂û ∂θ (ξ,ω0) =Ĝ(ξ,ω0). (4.12) Int. J. Anal. Appl. 18 (4) (2020) 684 Let V̂(ξ) denote the operator of the problem (4.12) for the Neumann-Neumann conditions (NN) and V̂(ξ) : W 2, 2(0,ω0) → L2(0,ω0) × C × C. If ξ is no eigenvalue of V̂(ξ), then for any F̂ ∈ L2(0,ω0) a unique weak solution û of (4.12) exists. We write û = V̂−1(ξ)[F̂,Ĝ(0),Ĝ(ω0)], (4.13) where V̂−1(ξ) represent the inverse (or resolvent) operator and V̂−1(ξ) : L2(0,ω0) × C × C → W 2, 2(0,ω0). Besides, the inverse Fourier transform of û yields the solution ũ(τ,θ) = u(x,y) of (3.2) and the subsequent regularity result holds. If no eigenvalues of V̂(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 and ũh(τ, θ) = uh(x) is the unique solution of (4.12) from W2, 2α (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 u(x) = uγ(x) = (2π) −1 2 ∫ ∞+iγ −∞+iγ eiξτ V̂−1(ξ)[F̂,Ĝ(0),Ĝ(ω0)] dξ. (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 ω0 > π, the following expansion of the solution of the boundary value (3.2) is obtained u(x) = C1 + C2 e πτ ω0 cos π ω0 θ + w(x), (4.15) where w(x) ∈ W2, 20 (S) and u(x) ∈ W 1, 2 0 (S) is the solution of the boundary value (3.2). Now, substituting r = eτ , we get u(x) = C1 + C2 r π ω0 cos π ω0 θ + w(x). (4.16) Mixed boundary conditions (ND) Let Pi be a boundary point at which the Neumann-Dirichlet conditions (ND) appear. Using the same approach which is used for the latter cases, i.e., (Dirichlet and Neumann) conditions, the following Fourier transformed form of (3.2) is obtained ξ2û− ∂2û ∂θ2 =F̂(ξ,θ), ∂û ∂θ (ξ, 0) =Ĝ(ξ, 0), û(ξ,ω0) =0. (4.17) Int. J. Anal. Appl. 18 (4) (2020) 685 Let V̂(ξ) denote the operator of the problem (4.17) and V̂(ξ) : W 2, 2(0,ω0) → L2(0,ω0) × C × C. If ξ is no eigenvalue of the operator V̂(ξ), then for any F̂ ∈ L2(0,ω0) a unique weak solution û of (4.17) exists. We can write û = V̂−1(ξ)[F̂,Ĝ(0), 0)], (4.18) where V̂−1(ξ) represent the resolvent operator and V̂−1(ξ) : L2(0,ω0) × C × C → W 2, 2(0,ω0). Further, the inverse Fourier transform of û yields the solution ũ(τ,θ) = u(x,y) of (3.2) and the subsequent regularity result holds. Again we have, if no eigenvalues of V̂(ξ) lie on the line h = Im ξ = α− 1, α ≥ 0, then the inverse Fourier transform which can be read as follows ũh(τ, θ) = (2π) −1 2 ∫ ∞+ih −∞+ih eiξτ V̂−1(ξ)[F̂,Ĝ(0), 0] dξ = uh(x) ∈W2, 2α (S), (4.19) exists and uh(x) is the uniquely determined solution from W2, 2α (S) of (3.2) for mixed conditions (ND). From [22, 23], a regularity result yields that u ∈ W2, 2γ+1(S) where γ is a sufficiently small positive real number. To derive an expansion of the solution u(x) in S, the main question is the inverse Fourier transformation of the right-hand sides of (4.17) which can be read as u(x) = uγ(x) = (2π) −1 2 ∫ ∞+iγ −∞+iγ eiξτ V̂−1(ξ)[F̂,Ĝ(0), 0] dξ. (4.20) The integral (4.20) can be calculated using the Cauchy theorem as same in (4.4) and the approach used for calculating the regularity results of the latter conditions. Moreover, the rectangle choosing here have the corner points −n+iγ, −n−i, n−i, n+iγ. Since, we have the eigenvalues ξl = i(l + 12 ) π ω0 , l = 0,±1,±2, ...,. For ω0 ∈ (π2 , 3π 2 ), we have ξ−1 = ( −i 2 ) π ω0 and ω0 > 3π 2 yield ξ−2 = ( −3i 2 ) π ω0 . Let these eigenvalues lie in the rectangle and the following expansion of the solution of the boundary value (3.2) is obtained u(x) = C1 e πτ 2ω0 cos π 2ω0 θ + w(x), for ω0 ∈ ( π 2 , 3π 2 ), u(x) = C1 e πτ 2ω0 cos π 2ω0 θ + C2 e 3πτ 2ω0 cos 3π 2ω0 θ + w(x), for ω0 > 3π 2 , (4.21) where w(x) ∈ W2, 20 (S) and u(x) ∈ W 1, 2 0 (S) is the solution of the boundary value (3.2). Now, substituting r = eτ , we get u(x) = C1 r π 2ω0 cos π 2ω0 θ + w(x), for ω0 ∈ ( π 2 , 3π 2 ), u(x) = C1 r π 2ω0 cos π 2ω0 θ + C2 r 3π 2ω0 cos 3π 2ω0 θ + w(x), for ω0 > 3π 2 . (4.22) Remark 4.1. It is observed that if the versed boundary conditions are used which means that the Dirichlet condition is at θ = 0 and the Neumann condition is at θ = ω0, then the similar regularity results can be Int. J. Anal. Appl. 18 (4) (2020) 686 obtained like the mixed conditions (ND) but the eigenvalues and the corresponding eigenfunctions discussed in Remark 3.2 are used. 4.1. The regularity of the boundary value problem in a polygonal domain. In Section 1, we have described that D is a polygonal domain and N denote the set of the boundary points which consists of{ P1, ...,PN } ⊂ ∂D. To investigate the regularity of the solution v of the boundary value problem (1.1) in D, the following set of boundary points of N are considered. Let we denote (1) J1 be the corresponding index set for the boundary points with the Dirichlet-Dirichlet boundary conditions (DD), where ωi > π, (2) J2 be the corresponding index set for the boundary points with the Neumann-Neumann boundary conditions (NN), where ωi > π, (3) J1J2 be the corresponding index set for the boundary points with the Neumann-Dirichlet boundary conditions (ND), where ωi ∈ (π2 , 3π 2 ), (4) J2J1 be the corresponding index set for the boundary points with the Neumann-Dirichlet boundary conditions (ND), where ωi > 3π 2 . Let v ∈ W 1,2(D) be the uniquely determined weak solution of (1.1). We investigate its regularity which can be written in the form v = ∑ i∈J1 χ2iv + ∑ i∈J2 χ2iv + ∑ i∈J1J2 χ2iv + ∑ i∈J2J1 χ2iv + ( 1 − ∑ i∈J1∪J2∪J1J2∪J2J1 χ2iv ) , (4.23) where the function χ is defined in (3.1) and u = χv. Using the expansions (4.11), (4.16) and (4.22) into (4.23), we get v = ∑ i∈J1 χiCi r π ωi i sin π ωi θi + ∑ i∈J2 χiCi r π ωi i cos π ωi θi + ∑ i∈J1J2 χiCi r π 2ωi i cos π 2ωi θi + ∑ i∈J2J1 ( χiCi r π 2ωi i cos π 2ωi θi + Ćiχi r 3π 2ωi i cos 3π 2ωi θi ) + w(x), (4.24) where w(x) ∈ W 2,2(D). Let θi represents the locally variable angle, i.e., 0 < θi < ωi, whereas Ci, Ći are the singularity coefficients and their computations can be found in [7, 8, 20]. Finally, (4.24) completely describe the regularity of the solution v of the boundary value problem (1.1). 5. Conclusion It is well-known from the theory of elliptic boundary value problems in domains with boundary irregular- ities, 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 Int. J. Anal. Appl. 18 (4) (2020) 687 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 (ωi > 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). 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