# Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations

Authors: Amila J. Maldeniya; Naleen C. Ganegoda; Kaushika De Silva; Sanath K. Boralugoda
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

---

## Abstract

In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test functionDℝ2. Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for Poisson’s equation in the upper half space ℝ+3 with Dirichlet boundary condition.


---

## Body

## 1. Introduction

In this paper, we find an integrable distributional solution for Poisson’s equation in the upper half space. Poisson’s equation is a second-order elliptic partial differential equation which has many applications in physics and engineering. There are three types of boundary conditions for Poisson’s equation: Dirichlet condition, Neumann conditions, and Robin conditions. This article is mainly concerning on Poisson’s equation with Dirichlet boundary condition (1):(1)△u=f,inΩ⊂ℝn,u=g,on∂Ω.Partial differential equations (PDEs) are more difficult to solve than ordinary differential equations (ODEs). Therefore, numerical approximations are widely used in application. There are standard numerical methods such as finite difference and finite element etc. to solve (1). In numerical approaches to solving PDEs, particularly in finite difference methods, we may face two ways of accuracy loss because of discretization of domain and approximating partial derivatives by difference formulas. Finite element methods will improve the accuracy of problems with irregular boundaries, but still abide by approximations as one has to do minimizing functionals with smaller class of functions [1]. Also, artificial neural network methods can be used for approximate nonsmooth solutions. See [2] for application of artificial neural network methods for Poisson’s equation with boundary value problems in domain ℝ2. This method trains data to optimize the algorithm which gives numerical solutions to partial differential equations. Convergent properties and accuracy of solution are not discussed in the given method of [2]. Finite element method can be used to solve elliptic problems with Henstock–Kurzweil integrable functions. Such an approximation is used in [3] for ODEs, where existence and uniqueness of the solution are not discussed. The proposed analytical approach in our paper will direct to a solution, where if its explicit form is not available, one has to only do a numerical integration.There are several methods to find analytical solution for Dirichlet problem such as Green’s function, Dirichlet’s principle, layer potentials, energy methods, and Perron’s method. In 1850 [4], George Green gives theoretical approach for Dirichlet problem which is now called as Green’s function. This assumes that Green’s function exists for any domain which is not true. However, this idea influences modern techniques and distributional theory. Using Green’s function, one can obtain a general representation solution for Poisson’s equation.General representation solution depends on the domainΩ and nature of the functions f and g. For Lebesgue integration, it is required that the conditions g∈C∂Ω and f∈CΩ get a unique analytical solution u∈C2Ω∩CΩ¯ for (1), where Ω needs to be open and bounded in ℝn ([5], pg. 28). In classical solution, it is worth to use Henstock–Kurzweil integral (HK-integral) instead of the Lebesgue integral.HK-integral is more advanced, and it includes the Lebesgue and Riemann integral. This integral was first introduced by Henstock and Kurzweil in 1957. It has many advantages such as convergence theorems, integration on unbounded intervals/functions, Fubini’s theorem, and fundamental theorem of calculus with full generality, for details see [6–9]. If ℋK is the space of HK-integrable functions, then it has proper inclusion L1⊊  ℋK. For an example, a highly oscillating function fx=1/xsin1/x3 is neither Lebesgue nor Riemann integrable. But f is HK-integrable. There are only few articles which use HK-integration to find analytical solution for Dirichlet problem [10, 11]. Talvila uses HK-integration to find solution for Dirichlet problem in the upper half plane, see example in [10]. Also, the same author obtains useful results of Poisson kernel in the unit disk via HK-integral with application into Dirichlet problem [11]. However, both articles [10, 11] prove its results in ℝ2 and do not extend into ℝ3. It is an obvious fact that properties of HK-integration are not easily extended for higher dimension. Even the HK-integration is superior to Lebesgue integration, there are a few drawbacks. Most significantly, there is no known natural topology for ℋK, so that it is not a Banach space. Therefore, there are not much functional analysis tools to work with the problem. However, there is a natural seminorm on ℋK defined by Alexiewicz [12].Looking at an alternative, we moved into distribution theory; specially the space of integrable distributionAC which is isomorphic to completion of the space ℋK, which is presented in Section 2. It seems the concept of integrable distribution is first introduced by Mikusiński and Ostaszewski [13] and further developed by several authors [14, 15]. In recent years, integrable distributional solution for differential equations has been used by many authors [16–19]. We define derivative (in distributional sense) of AC and the integral on it. In Section 3, the space of Hardy–Krause variation functions is identified as a multiplier for ℋK. Here it uses Henstock–Stieltjes integral and integration by parts defined on AC. In [15], example 14.4, Talvila discussed the Dirichlet boundary condition of Laplace equation in the half space. There it uses Poisson Kernel, and proof of integrable distributional solution for the Laplace equation is not given. We extend the proof of this problem in Section 4 and obtain integrable distributional solution for (1) when f=0.


## 2. Integrable Distributions

Distribution theory was frequently used with the advent of Laurent Schwartz [20] in 1950. Distributions or generalized functions are an important functional analysis tool in modern analysis, especially in the field of PDE’s. When functions are nonsmooth, then distribution theory allows to perform operations: translation, differentiation, convolution etc. Here, we use integrable distributions which are a special class of distributions that will be important for applications of distribution theory to partial differential equations, for details see [13, 14, 21]. There are several articles on solving ordinary and partial differential equations using integrable distributions [16, 17, 19]. Integrable distribution has used to find a distributional solution for Poisson’s equation with Dirichlet boundary condition for the upper half plane. This article extends the distributional solution for Poisson’s equation for the 3-dimensional upper half space.Let the space of test functions isDℝ2=ϕ:ℝ2⟶ℝ|ϕ∈C∞ and ϕ has compact support}. Then, Dℝ2 forms a normed space under usual pointwise operations. The dual space, D′ℝ, is called as the space of distributions. That is, the distributions are the continuous linear functionals on Dℝ2. For a given distribution T, denotes its action on test function ϕ∈Dℝ2 by T,ϕ. For an example, if f is a locally HK-integrable function, then its corresponding distribution is defined Tf,ϕ=∫−∞∞∫−∞∞fϕ. Hereafter, all integral will be HK-integral unless stated otherwise. Next, consider derivative of a distribution, sometimes called as distributional derivative or weak derivative. If F is continuously differentiable, then ∫−∞∞F′ϕ=−∫−∞∞Fϕ′ for any test function ϕ. The compact support of ϕ was used in this integration by parts. With this suggestion, we define the distributional derivative of a distribution F∈D′ℝ by ∂αF,ϕ=−1αF,∂αϕ for any ϕ∈Dℝ2. Therefore, in particular, ∂12F,ϕ=F,∂12ϕ, where ∂12=∂2/∂y∂x. Now, consider a specific type of distributions called as integrable distributions. This definition seems to have been first introduced by Mikusiński and Ostaszewski [13]. After that, it was developed in detail in the plane by Ang et al. [14] and on ℝ¯ and ℝ¯2 by Talvila [15, 21].For the extended real numbersℝ¯, consider C0ℝ¯2 be the space of continuous functions so that both limx⟶−∞Fx,y and limy⟶−∞Fx,y exit for any F∈Cℝ¯2. Then, define ℬCℝ¯2=F∈C0ℝ¯2|F−∞,y=Fx,−∞=0,∀x,y∈ℝ¯. Then, ℬCℝ¯2 forms a Banach space under the uniform norm ⋅∞. Next, define the integrable distribution as the derivative of function in ℬCℝ¯2.Definition 1.
Letf∈D′ℝ2 be distribution. The space of integrable distributions is defined and denoted by(2)ACℝ¯2=f∈D′ℝ2|f=∂12F,F∈ℬCℝ¯2.Here,f=∂12F be the sense of distributional derivative, i.e., f,ϕ=∂12F,ϕ=F,∂12ϕ for ϕ∈Dℝ2. The function F is called as a primitive of f. Generally, primitive is not unique and differed by a constant. However, with our choice F−∞,y=Fx,−∞=0, primitive is unique for any given f∈D′ℝ2.Theorem 1.
Any givenf∈ACℝ¯2 has a unique primitive in ℬCℝ¯2.Proof.
LetF1 and F2 are primitives of f∈ACℝ¯2. Then, f=∂12F1=∂12F2 so that ∂12F1−F2=0. This implies F1−F2=Xx+Yy for some X,Y∈Cℝ¯2. Since F1−F2∈ℬCℝ¯2, for a fixed y0(3)0=limx⟶−∞F1−F2=limx⟶−∞Xx+limx⟶−∞Yy0.
Therefore,limx⟶−∞Xx=−Yy0 for any x∈ℝ. This implies that Xx is a constant function. For fixed x0, letting y⟶−∞ gives Yy is a constant function for the same argument. Those constant functions sums into 0, and hence F1=F2.The uppercase letter is denoted the primitive of any integrable distributionf. If f∈ACℝ¯2 with the primitive F∈ℬCℝ¯2, then for any test function ϕ∈Dℝ2,(4)f,ϕ=∂12F,ϕ=F,∂12ϕ=∫−∞∞∫−∞∞F∂12ϕ,where it uses HK-integral, and here onward the same, unless otherwise stated. Since the linearity of derivatives, ACℝ¯2 forms a vector space with usual operations of functionals. To get Banach space structure of ACℝ¯2, it requires to define the suitable norm. Using the unique primitive of f∈ACℝ¯2, we define the Alexiewicz norm as(5)f=sup∫−∞x∫−∞yf:x,y∈ℝ2=supℝ2F=F∞.Note that the spaceℋK with the Alexiewicz norm is a normed space, but not a Banach space [12, 22]. Absence of Banach space structure of ℋK is main disadvantage on it. It is one of the reasons to move into ACℝ¯2.Theorem 2.
ACℝ¯2 is a Banach space with the Alexiewicz norm ⋅.Proof.
First, prove⋅ is a norm.(i)
Iff≡0, then clearly f=0. If f=0 and then F∞=0. This implies F≡0 and hence f=∂12F≡0.(ii)
For anyλ∈ℝ and f∈ACℝ¯2,(6)λf=sup∫−∞x∫−∞yλf:x,y∈ℝ2=λsup∫−∞x∫−∞yf:x,y∈ℝ2=λf.(iii)
Iff,g∈ACℝ¯2, the primitive of f+g is F+G. Then,(7)f+g=F+G∞≤F∞+G∞=f+g.
Therefore,ACℝ¯2 is a normed space. To show completeness, let any fn⊆ACℝ¯2 be Cauchy sequence in ⋅. Then, Fn⊆ℬCℝ¯2 is a Cauchy sequence in ⋅∞ because fn−fm=Fn−Fm∞. But ℬCℝ¯2 is a Banach space, so that there exists a F∈ℬCℝ¯2 and Fn⟶F in ⋅∞. Let f=∂12F. Then, f∈ACℝ¯2 and fn−f=Fn−F∞⟶0. This shows fn⟶f∈ACℝ¯2 in ⋅ and hence the lemma.Alexiewicz norm dose not induce an inner product inACℝ¯2 because the norm does not satisfy the parallelogram law. Therefore, ACℝ¯2 is not a Hilbert space under the norm ⋅. However, the space ACℝ¯2 with ⋅ is isomorphic to ℬCℝ¯2 with ⋅∞. The mapping f⟶F gives a linear isomorphism between the spaces. Uniqueness of primitive gives the mapping is injective, and by the definition it is surjective mapping. Furthermore, the mapping preserves norm since f=F∞. Thus, the space ACℝ¯2 is identified with the space of continuous functions vanish at infinity.Proposition 1.
ACℝ¯2,⋅ is isometrically isomorphic to ℬCℝ¯2,⋅∞.Next, we define the integral of an integrable distribution, as it uses the primitiveF∈ℬCℝ¯2 of f∈ACℝ¯2.Definition 2.
(see [15], Definition 4.3) Let f∈ACℝ¯2 with the primitive F∈ℬCℝ¯2. We define integral of f on I=a,b×c,d⊆ℝ¯2 by(8)∫If=∫ab∫cdf=Fa,c+Fb,d−Fa,d−Fb,c.This integral is uniquely defined by uniqueness of the primitive, Theorem1. Also, it is very general and includes the Riemann integral, Lebesgue integral, and HK-integral. Norm preservation of Proposition 1 gives that the corresponding integral values agree on each integral as presented in Theorem 3.Theorem 3.
For any test functionϕ∈Dℝ2, if f∈L1ℝ2 is identity with the corresponding distribution f:ϕ⟼L∫ℝ2fϕ (or with HK-integral), then(i)
f∈ACℝ¯2(ii)
ACℝ¯2 is the completion of L1ℝ2 (or ℋKℝ2) with respect to the norm:(9)f=supL∫−∞x∫−∞yf:x,y∈ℝ2.(or in the sense of HK-integral)(iii)
∫ℝ2f=L∫ℝ2for=HK∫ℝ2fProof.
(i)
Letf∈L1ℝ2. For any x,y∈ℝ¯2, we define FLx,y=L∫−∞x∫−∞yf. Then, FL is a primitive of f in the Lebesgue integral (or HK-integral). Then, FL∈C0ℝ2 since the primitive of Lebesgue integral (or HK-integral) is continuous. Clearly, FL−∞,y=FLx,−∞=0 for any x,y∈ℝ¯2. Therefore, f∈ACℝ¯2.(ii)
From (i),L1ℝ2⊂ACℝ¯2. Let any f∈ACℝ¯2. Then, it needs to show f∈L1ℝ2¯. We prove this by showing f is a limit point of L1ℝ2¯. Since f∈ACℝ¯2 there is F∈ℬCℝ¯2. Now choose a sequence Fn⊂C2ℝ¯2 so that Fn−F∞⟶0. Since Fn converges to F uniformly and F−∞,y=Fx,−∞=0, we can assume that Fn−∞,y=Fnx,−∞=0 for each n. Now, take fn=∂Fn in the distributional sense. Then, fn=∂Fn∈Cℝ2⊂L1ℝ2. Moreover, fn−f=Fn−F∞⟶0. Hence, f is a limit point of L1ℝ2¯.(iii)
Any Lebesgue integrable function is HK-integrable with the same value. Thus, the result.Note that iff∈L1ℝ2, then its Alexiewicz norm and L1 norm are not equivalent, for an example see [15] (pg. 9). The norms are equivalent if f≥0 for almost everywhere. From the definition of integral of f∈ACℝ¯2 and its unique primitive, we obtain following fundamental theorem of calculus.Theorem 4.
(fundamental theorem of calculus).(i)
IfF∈Cℝ2 then ∫−∞x∫−∞y∂F=Fx,y+F∞,∞−F−∞,y−Fx,−∞(ii)
Letf∈ACℝ¯2 and Fx,y=∫−∞x∫−∞yf. Then F∈ℬCℝ¯2 and f=∂FProof.
See [15].


## 3. Multiplier for Integrable Distributions

We will turn now to our discussion on multipliers forACℝ¯2. A multiplier for a class of functions ℱ is a function g such that fg∈ℱ for each f∈ℱ. For an example, in Lebesgue integral, g∈L∞ℝn if and only if fg∈L1ℝn for each f∈L1ℝn. i.e., the space L∞ℝn is multipliers for L1ℝn. For HK-integral in one-dimensional case 2, space of bounded variation functions, ℬV, is multipliers for ℋK [9], (Theorem 6.1.5). In multidimension case, there is no unique way to extend the notion of variation to function. Adams and Clarkson [23] give six such extensions but mostly recognized by Vitali variation and Hardy–Krause variation. To establish integration by parts formula and multipliers for ACℝ¯2, we begin with Hardy–Krause variation.Two intervals inℝ¯2 are nonoverlapping if their intersection is of Lebesgue measure zero. A division of ℝ¯2 is a finite collection of nonoverlapping intervals whose union is ℝ¯2. If φ:ℝ¯2⟶ℝ, then its total variation (in the sense of Vitali) is given by(10)V12φ=supD∑iφai,ci+φbi,di−φai,di−φbi,ci,where the supremum is taken over all divisions D of ℝ¯2 and interval Ii=ai,bi×ci,di∈D.Definition 3.
(Hardy–Krause bounded variation). A functionφ:ℝ¯2⟶ℝ is said to be Hardy–Krause bounded variation if the following conditions are satisfied.(i)
V12φ is finite(ii)
The functionx⟼φx,y0 is bounded variation on ℝ¯ for some y0∈ℝ¯(iii)
The functiony⟼φx0,y is bounded variation on ℝ¯ for some x0∈ℝ¯The set of functions in Hardy–Krause variation is denoted byℬVHKℝ¯2. As in [15], ℬVHKℝ¯2 forms a Banach space with norm defined by φℬV=φ∞+V1φ∞+V2φ∞+V12φ. Here, V1φ and V2φ are bounded variation to the functions of (ii) and (iii) in Definition 3. For any multidimensional case, any function of Hardy–Krause variation is a multiplier for HK-integrable functions, see [9]. Therefore, define following integration by parts, as a two-dimensional case which is presented in ([9], Theorem 6.5.9).Definition 4.
(integration by parts). Letf∈ACℝ¯2 with its primitive F∈ℬCℝ¯2. If g∈ℬVℋKℝ¯2, then define integration by parts on a,b×c,d⊆ℝ¯2.(11)∫ab∫cdfg=Fa,cga,c+Fb,dgb,d−Fa,dga,d−Fb,cgb,c−∫abFx,dd1gx,d+∫abFx,cd1gx,c−∫cdFb,yd2gb,y+∫cdFa,yd2ga,y+∫ab∫cdFx,yd12gx,y.Here,dgx,y is Henstock–Stieltjes integral of relevant variables, see in detail [24]. This integration by parts formula induces the space of Hardy–Krause bounded variation functions, ℬVℋKℝ¯2, as a multiplier for ACℝ¯2. This multiplier is important to define convolution and obtain convergence theorem.It needs to obtain continuity of the Alexiewicz norm since the Dirichlet boundary condition is taken on it. For anys,t∈ℝ2, the translation of f∈ACℝ¯2⊆D′ℝ2 is defined by τs,tf,ϕ=f,τ−s,−tϕ for ϕ∈Dℝ2, where τs,tϕx,y=ϕx−s,y−t. Translation is invariant and continuous under the Alexiewicz norm.Theorem 5.
Letf∈ACℝ¯2 and s,t∈ℝ2. Then, (i) τs,tf∈ACℝ¯2, (ii) f=τs,tf, and (iii) f−τs,tf⟶0 as s,t⟶0,0. i.e., continuity of f in the Alexiewicz norm.Proof.
LetF∈ℬCℝ¯2 be the primitive of f∈ACℝ¯2. Then, τs,tF is a continuous function.(i)
For anyϕ∈Dℝ2, applying change of variable in integral,(12)τs,tf,ϕ=f,τ−s,−tϕ=∫ℝ∫ℝfx,yϕx+s,y+tdxdy=∫ℝ∫ℝfx−s,y−tϕx,ydxdy=∫ℝ∫ℝ∂12Fx−s,y−tϕx,ydxdy=∫ℝ∫ℝ∂12τs,tFx,yϕx,ydxdy.Therefore,τs,tF∈ℬCℝ¯2 is the primitive of τs,tf.(ii)
Clearly,F∞=τs,tF∞ for any s,t∈ℝ2. Therefore, f=τs,tf.(iii)
Iff∈ACℝ¯2, then f−τs,tf∈ACℝ¯2 so that(13)f−τs,tf=F−τs,tF∞=supx,y∈ℝ2Fx,y−Fx−s,y−t⟶0,when s,t⟶0,0 because of uniform continuity of F.


## 4. Integrable Distributional Solution for Poisson Equation

Poisson’s kernel and its convolution are used to find a classical solution to the Dirichlet problem. In this section, we use convolution and convergence theorem onACℝ¯2 to find integrable distributional solution. If g∈ℬVℋKℝ¯2, then τ−s,−tg∈ℬVℋKℝ¯2. From integration by parts, the convolution(14)f∗gx,y=∫ℝ∫ℝfτ−s,−tgdsdt=∫ℝ∫ℝfx−s,y−tgx,ydsdtexists for f∈ACℝ¯2 and g∈ℬVℋKℝ¯2. Proposition 2 gives some basic properties of convolution which help with our problem.Proposition 2.
Iff∈ACℝ¯2 and g∈ℬVℋKℝ¯2, then (i) f∗g exists on ℝ2 and (ii) f∗g=g∗f.Proof.
See [15], Theorem 14.1.Convergence theorems on Banach space are important when solving partial differential equations. There are several convergence theorems inAC such as strong convergence in AC, weak/strong convergence in ℬV, and weak/strong convergence in D. Details are discussed in the plane by Ang, Schmitt and Vy [14] and on ℝ¯ by Talvila [2]. Now, we move into the convergence theorem on ACℝ¯2 which will help to interchange limit of integral.Proposition 3.
Letφn be a sequence in ℬVℋKℝ¯2 so that φnℬV<∞ and limn⟶∞φn=φ pointwise on ℝ¯2 for a function φ. If g∈ACℝ¯2, then g∈ℬVℋKℝ¯2 and(15)limn⟶∞∫ℝ∫ℝgφn=∫ℝ∫ℝgφ.Proof.
See [15], Proposition 9.1.Let us move into the main problem. As in [25] (pg. 84), Poisson’s equation with Dirichlet boundary condition is easily reduced to the case where either f=0 or g=0 in (1). Therefore, consider integrable distributional solution for the following problem:(16)△u=0,inℝ+3,u=g,on∂ℝ+3,where ℝ+3=x,y,z∈ℝ3:z>0. For z>0, consider Poisson’s kernel for the upper half space in 3-dimension(17)Kx,y,z=zx2+y2+z2−3/22π.Note that∫ℝ∫ℝKdxdy=1. If g∈Cℝ2∩L∞ℝ2, then classical solution for the problem is given by the following convolution:(18)ux,y,z=Kx,y,.∗gx,y=∫ℝ∫ℝKx−s,y−t,.gs,tdtds.This convolution gives integrable distributional solution for (16).Theorem 6.
Letg∈ACℝ¯2 and define ux,y,z by (18). Then,(i)
For eachx,y∈ℝ and z>0, the convolution Kx,y,.∗gx,y=ux,y,z exists and u∈ACℝ¯2(ii)
△ux,y,z=0 in ℝ+3(iii)
limz⟶0+ux,y,z−gx,y=0, for x,y∈ℝProof.
(i)
Fixx,y∈ℝ and z>0. Then, the function s,t⟼Kx−s,y−t,z is of Hardy–Krause variation on ℝ2. From Proposition 2, K∗g exists on the upper half space. Also, u=K∗g∈ACℝ¯2 because any function of Hardy–Krause variation is a multiplier for HK-integrable functions.(ii)
Fixx,y∈ℝ and z>0. To get ux, let xn⟶x be a sequence so that xn≠x. For any s,t∈ℝ2, define(19)φns,t=Kxn−s,y−t,z−Kx−s,y−t,zxn−x.Then,ux=limn∫ℝ∫ℝgs,tφns,tdsdt. Since Kx−s,y−t,z of Hardy–Krause bounded variation, so does φns,t for each n. Since g∈ACℝ¯2, from Proposition 3(20)ux=∫ℝ∫ℝlimn⟶∞gs,tφns,tdsdt=∫ℝ∫ℝgs,tKxx−s,y−t,zdsdt.Repeat the same argument to obtainuxx,uy,uyy,uz, and uzz. From the linearity of integral,(21)△ux,y,z=∫ℝ∫ℝgs,t△Kx−s,y−t,zdsdt.But Poisson kernel is harmonic, so that△ux,y,z=0 in ℝ+3.(iii)
Let anyα,β∈ℝ. For Poisson kernel ∫ℝ∫ℝKdxdy=1. Using Fubini–Tonelli theorem and interchanging iterated integral,(22)∫−∞α∫−∞βu−gdxdy=∫−∞α∫−∞βux,y,z−gx,ydxdy=∫−∞α∫−∞βgx,y∗Kx,y,z−gx,ydxdy=∫−∞α∫−∞β∫ℝ∫ℝKs,t,zgx−s,y−t−gx,ydsdtdxdy=∫−∞α∫−∞β∫ℝ∫ℝz2πs2+t2+z23/2gx−s,y−t−gx,ydsdtdxdy=∫−∞α∫−∞β∫ℝ∫ℝ12πs2+t2+13/2gx−zs,y−zt−gx,ydsdtdxdy=∫ℝ∫ℝ12πs2+t2+13/2∫−∞α∫−∞βgx−zs,y−zt−gx,ydxdydsdt≤∫ℝ∫ℝKs,t,1gx−zs,y−zt−gx,ydsdt≤2g.
Sinceα and β are arbitrary,(23)ux,y,z−gx,y≤∫ℝ∫ℝKs,t,1gx−zs,y−zt−gx,ydsdt≤2g.
To letz⟶0+, apply dominated convergence theorem (HK-integral) for the sequence of functions:(24)gns,t=Ks,t,1gx−1ns,y−1nt−gx,y.
Then,gns,t≤2gKs,t,1∈ℋKℝ2 since ∫ℝ∫ℝKs,t,1dsdt=1. It gives(25)limn⟶∞∫ℝ∫ℝgns,tdsdt=limn⟶∞∫∫ℝKs,t,1gx−zs,y−zt−gx,ydsdt=∫ℝ∫ℝlimn⟶∞gns,tdsdt=∫ℝ∫ℝKs,t,1limn⟶∞gx−1ns,y−1nt−gx,ydsdt.
From part (iii) of Theorem5, we get limn⟶∞gx−1/ns,y−1/nt−gx,y=0 so that limit of (23), limz⟶0+ux,y,z−gx,y=0, and this completes the theorem.Next, we give an example which will apply Theorem6.Example 1.
Consider the problem (16) in which(26)gx,y=sinxsinyxy,x≠0andy≠0,0,x=0ory=0.
Then,g is HK-integrable but not absolutely integrable. Hence, g∈ACℝ¯2∖L1ℝ2. From Theorem 6,(27)ux,y,z=∫ℝ∫ℝKx−s,y−t,.gs,tdtds=∫ℝ∫ℝz2πx−s2+y−t2+z2−3/2sinssintstdtds,exists and gives solution for the example. Note that Kg∈ℋKℝ2 since K∈ℬVℋKℝ¯2 is a multiplier for ℋKℝ2. However, the solution does not exist with the Lebesgue integral. For contraposition, suppose that Kg∈L1ℝ2. Then, it follows from Fubini’s theorem [9] (Theorem 2.5.5) that the function(28)h:s⟼Kx−s,y−t,zgs,t,belongs to L1ℝ1 for almost all t∈ℝ. That is,(29)hs=zsint2πtsinssx−s2+y−t2+z23/2∈L1ℝ1=h0ssinss,where h0s=zsint/2πt1/x−s2+y−t2+z23/2. This contradicts that hs∉L1ℝ1. To see that, suppose hs∈L10,1⊂L1ℝ1. Since h0s∈L∞0,1 is multiplier for L10,1, it follows that sins/s∈L10,1. This implies that sins/s absolutely integrable on 0,1, which is a contradiction. Hence, hs∉L1ℝ1.Even HK-integral exists in (27), it may not be able to evaluate and get explicit function. In that case, numerical integration can be applied. Numerical integration should be done with respect to s and t while fixing x,y, and z, details in [26]. This reduces solving the partial differential equation into applying numerical integration for the double integral (27).


## 5. Discussion

In this work, integrable distributional solution is obtained for Poisson equation with the Dirichlet boundary condition in the upper half space. Boundary conditions are taken in the Alexiewicz norm. Lebesgue integral uses for the classical solution to Poisson equation. Because of the proper inclusionL1⊊  ℋR, we may use HK-integral to solve Poisson equation. However, there are some limitations due to the absence of any natural topology. That is, ℋK is not a Banach space; nevertheless, there is a natural seminorm called Alexiewicz. This is one of the main reasons to move into the space of integrable distribution ACℝ¯2, which contains ℋK and leads to a Banach space with Alexiewicz norm.Integrable distribution is defined as derivative (distributional sense) of function inℬCℝ¯2. In this definition, any f∈ACℝ¯2 identifies with the unique primitive F∈ℬCℝ¯2. This unique primitive allows to define a Alexiewicz norm ⋅ on ACℝ¯2 and leads to a Banach space. Alexiewicz norm does not induce an inner product in ACℝ¯2 because it does not satisfy the parallelogram law. Therefore, ACℝ¯2 is not a Hilbert space under the norm ⋅. However, the space ACℝ¯2,⋅ is isometrically isomorphic to the space ℬCℝ¯2,⋅∞. Integral on ACℝ¯2 is uniquely defined by uniqueness of the primitive. ACℝ¯2 is the completion of L1ℝ2 (or ℋKℝ2) with respect to the corresponding Alexiewicz norm.In one-dimensional HK-integral, space of bounded variation functions,ℬV, is multipliers for ℋK. In multidimension case, there are several extensions to variation of function. To establish integration by parts formula and multipliers for ACℝ¯2, we begin with Hardy–Krause variation. The space of Hardy–Krause bounded variation functions, ℬVℋKℝ¯2, as a multiplier for ACℝ¯2. This multiplier is important to define convolution and obtain convergence theorem. ℬVℋKℝ¯2 is used to define convolution and obtain convergence theorem on ACℝ¯2. There are several convergence theorems in AC such as strong convergence in AC, weak/strong convergence in ℬV, and weak/strong convergence in D. Convergence theorem on ACℝ¯2 and dominated convergence theorem in ℋKℝ2 are used to obtain integrable distributional solution. Finding of this article gives solution for Poisson equation for broader initial condition in the upper half space. Dirichlet problem in ℝ3 is considered, and the solution has obtained by HK-integration where it is not possible from Lebesgue integral. Because of function’s complexity, it may not be able to have solution in explicit form. It would help keep an analytical outlook in the solution, which would be better than solving the partial differential equation by numerical methods such as finite difference methods and finite element methods.There is potential to find integrable distributional solution for Poisson equation in other boundary conditions, Neumann and Robin. But it needs to develop suitable convergence theorems onAC and its multipliers ℬVℋK. Also, it is possible to consider Poisson equation in unit ball with the relevant Poisson kernel. Talvila, in [15], gives some construction to develop integrable distribution on ℝn, integration by parts, and mean value theorem and defines ACℝ¯n and ℬCℝ¯n for multidimensional case. Hence, integrable distributional solution is possible to obtain for Poisson equation with Dirichlet boundary condition in ℝ+n with suitable convergence theorem.



---

*Source: 5511283-2021-07-12.xml*