Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2021, Article ID 5511283, 9 pages https://doi.org/10.1155/2021/5511283

Research Article Solving Poisson Equation by Distributional HK-: Prospects and Limitations

Amila J. Maldeniya ,1 Naleen C. Ganegoda ,2 Kaushika De Silva,2 and Sanath K. Boralugoda3

1Department of Mathematics, General Sir John Kotelawala Defence University, Kandawala Road, Ratmalana, Sri Lanka 2Department of Mathematics, University of Sri Jayewardenepura, Gangodawila, Nugegoda, Sri Lanka 3Department of Mathematics, University of Colombo, Cumarathunga Munidasa Mawatha, Colombo, Sri Lanka

Correspondence should be addressed to Amila J. Maldeniya; [email protected]

Received 11 February 2021; Revised 14 May 2021; Accepted 24 June 2021; Published 12 July 2021

Academic Editor: Ra´ul E. Curto

Copyright © 2021 Amila J. Maldeniya et al. &is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test function D(R2). Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for 3 Poisson’s equation in the upper half space R+ with Dirichlet boundary condition.

1. Introduction improve the accuracy of problems with irregular boundaries, but still abide by approximations as one has to do mini- In this paper, we find an integrable distributional solution mizing functionals with smaller class of functions [1]. Also, for Poisson’s equation in the upper half space. Poisson’s artificial neural network methods can be used for approx- equation is a second-order elliptic partial differential imate nonsmooth solutions. See [2] for application of ar- equation which has many applications in physics and en- tificial neural network methods for Poisson’s equation with gineering. &ere are three types of boundary conditions for boundary value problems in domain R2. &is method trains Poisson’s equation: Dirichlet condition, Neumann condi- data to optimize the algorithm which gives numerical so- tions, and Robin conditions. &is article is mainly con- lutions to partial differential equations. Convergent prop- cerning on Poisson’s equation with Dirichlet boundary erties and accuracy of solution are not discussed in the given condition (1): method of [2]. Finite element method can be used to solve

n elliptic problems with Henstock–Kurzweil integrable △u � f, in Ω ⊂ R , functions. Such an approximation is used in [3] for ODEs, � (1) u � g, on zΩ . where existence and uniqueness of the solution are not discussed. &e proposed analytical approach in our paper Partial differential equations (PDEs) are more difficult to will direct to a solution, where if its explicit form is not solve than ordinary differential equations (ODEs). &ere- available, one has to only do a numerical integration. fore, numerical approximations are widely used in appli- &ere are several methods to find analytical solution for cation. &ere are standard numerical methods such as finite Dirichlet problem such as Green’s function, Dirichlet’s difference and finite element etc. to solve (1). In numerical principle, layer potentials, energy methods, and Perron’s approaches to solving PDEs, particularly in finite difference method. In 1850 [4], George Green gives theoretical ap- methods, we may face two ways of accuracy loss because of proach for Dirichlet problem which is now called as Green’s discretization of domain and approximating partial deriv- function. &is assumes that Green’s function exists for any atives by difference formulas. Finite element methods will domain which is not true. However, this idea influences 2 International Journal of Mathematics and Mathematical Sciences modern techniques and distributional theory. Using Green’s equation is not given. We extend the proof of this problem in function, one can obtain a general representation solution Section 4 and obtain integrable distributional solution for (1) for Poisson’s equation. when f � 0. General representation solution depends on the domain Ω and nature of the functions f and g. For Lebesgue in- 2. Integrable Distributions tegration, it is required that the conditions g ∈ C(zΩ) and f ∈ C(Ω) get a unique analytical solution Distribution theory was frequently used with the advent of u ∈ C2(Ω) ∩ C(Ω) for (1), where Ω needs to be open and Laurent Schwartz [20] in 1950. Distributions or generalized bounded in Rn ([5], pg. 28). In classical solution, it is worth functions are an important functional analysis tool in to use Henstock–Kurzweil integral (HK-integral) instead of modern analysis, especially in the field of PDE’s. When the Lebesgue integral. functions are nonsmooth, then distribution theory allows to HK-integral is more advanced, and it includes the perform operations: translation, differentiation, convolution Lebesgue and Riemann integral. &is integral was first in- etc. Here, we use integrable distributions which are a special troduced by Henstock and Kurzweil in 1957. It has many class of distributions that will be important for applications advantages such as convergence theorems, integration on of distribution theory to partial differential equations, for unbounded intervals/functions, Fubini’s theorem, and details see [13, 14, 21]. &ere are several articles on solving fundamental theorem of calculus with full generality, for ordinary and partial differential equations using integrable details see [6–9]. If HK is the space of HK-integrable distributions [16, 17, 19]. Integrable distribution has used to functions, then it has proper inclusion L1 ⊊ HK. For an find a distributional solution for Poisson’s equation with example, a highly oscillating function f(x) � (1/x) Dirichlet boundary condition for the upper half plane. &is sin(1/x3) is neither Lebesgue nor Riemann integrable. But f article extends the distributional solution for Poisson’s is HK-integrable. &ere are only few articles which use HK- equation for the 3-dimensional upper half space. integration to find analytical solution for Dirichlet problem Let the space of test functions is [10, 11]. Talvila uses HK-integration to find solution for D(R2) � �ϕ: R2 ⟶ R|ϕ ∈ C∞ and ϕ has compact sup- Dirichlet problem in the upper half plane, see example in port}. &en, D(R2) forms a normed space under usual [10]. Also, the same author obtains useful results of Poisson pointwise operations. &e dual space, D′(R), is called as the kernel in the unit disk via HK-integral with application into space of distributions. &at is, the distributions are the Dirichlet problem [11]. However, both articles [10, 11] prove continuous linear functionals on D(R2). For a given dis- its results in R2 and do not extend into R3. It is an obvious tribution T, denotes its action on test function ϕ ∈ D(R2) by fact that properties of HK-integration are not easily ex- 〈T, ϕ〉. For an example, if f is a locally HK-integrable tended for higher dimension. Even the HK-integration is function, then its corresponding distribution is defined 〈T , 〉 � �∞ �∞ f superior to Lebesgue integration, there are a few drawbacks. f ϕ − ∞ − ∞ ϕ. Hereafter, all integral will be HK- Most significantly, there is no known natural topology for integral unless stated otherwise. Next, consider derivative of HK, so that it is not a Banach space. &erefore, there are not a distribution, sometimes called as distributional derivative or weak derivative. If F is continuously differentiable, then much functional analysis tools to work with the problem. ∞ ∞ � F′ � − � F ′ However, there is a natural on HK defined by − ∞ ϕ − ∞ ϕ for any test function ϕ. &e compact Alexiewicz [12]. support of ϕ was used in this integration by parts. With this Looking at an alternative, we moved into distribution suggestion, we define the distributional derivative of a F ′( ) 〈zαF, 〉 � (− )|α|〈F, zα 〉 theory; specially the space of integrable distribution AC distribution ∈ D R by ϕ 1 ϕ for which is isomorphic to completion of the space HK, which any ϕ ∈ D(R2). &erefore, in particular, 2 is presented in Section 2. It seems the concept of integrable 〈z12F, ϕ〉 � 〈F, z12ϕ〉, where z12 � z /zy zx. Now, consider distribution is first introduced by Mikusinski´ and Ostas- a specific type of distributions called as integrable distri- zewski [13] and further developed by several authors butions. &is definition seems to have been first introduced [14, 15]. In recent years, integrable distributional solution for by Mikusinski´ and Ostaszewski [13]. After that, it was de- veloped in detail in the plane by Ang et al. [14] and on R and differential equations has been used by many authors 2 A R by Talvila [15, 21]. [16–19]. We define derivative (in distributional sense) of C 2 and the integral on it. In Section 3, the space of Har- For the extended real numbers R, consider C0(R ) be dy–Krause variation functions is identified as a multiplier for the space of continuous functions so that both HK. Here it uses Henstock–Stieltjes integral and integra- limx⟶− ∞F(x, y) and limy⟶− ∞F(x, y) exit for any A 2 2 tion by parts defined on C. In [15], example 14.4, Talvila F ∈ C(R ). &en, define B (R 2) � �F ∈ C0(R ) |F(− ∞, discussed the Dirichlet boundary condition of Laplace C 2 equation in the half space. &ere it uses Poisson Kernel, and y) � F(x, − ∞) � 0, ∀x, y ∈ R}. &en, BC(R ) forms a proof of integrable distributional solution for the Laplace Banach space under the uniform ‖·‖∞. Next, define the International Journal of Mathematics and Mathematical Sciences 3 integrable distribution as the derivative of function in (i) If f ≡ 0, then clearly ‖f‖ � 0. If ‖f‖ � 0 and then 2 ‖F‖ � 0. &is implies F ≡ 0 and hence BC(R ). ∞ f � z12F ≡ 0. 2 Definition 1. Let f ∈ D′(R2) be distribution. &e space of (ii) For any λ ∈ R and f ∈ AC(R ), � � integrable distributions is defined and denoted by � x y � � � 2 ‖λf‖ � sup��� � λf�: (x, y) ∈ R � 2 2 2 − ∞ − ∞ AC�R � ��f ∈ D′�R �|f � z12F, F ∈ BC�R ��. (2) � x y � � � 2 �|λ|sup��� � f�: (x, y) ∈ R � �|λ|‖f‖. � − ∞ − ∞ � (6) Here, f � z12F be the sense of distributional derivative, i.e., 〈f, ϕ〉 � 〈z F, ϕ〉 � 〈F, z ϕ〉 for ϕ ∈ D(R2). &e 12 12 f, g ( 2) (f + g) function F is called as a primitive of f. Generally, primitive (iii) If ∈ AC R , the primitive of is (F + G) is not unique and differed by a constant. However, with our . &en, choice F(− ∞, y) � F(x, − ∞) � 0, primitive is unique for any given f ∈ D′(R2). ‖f + g‖ �‖F + G‖∞ ≤ ‖F‖∞ +‖G‖∞ �‖f‖ +‖g‖. (7) 2 2 Theorem 1. Any given f ∈ AC(R ) has a unique primitive A (R ) 2 &erefore, C is a normed space. To show com- in BC(R ). 2 pleteness, let any �fn � ⊆ AC(R ) be Cauchy sequence in ‖·‖. 2 2 &en, �F � ⊆ B (R ) is a Cauchy sequence in ‖·‖ because Proof. Let F and F are primitives of f ∈ A (R ). &en, n C ∞ 1 2 C ‖f − f ‖ � ‖F − F ‖ ( 2) f � z F � z F so that z (F − F ) � 0. &is implies n m n m ∞. But BC R is a Banach space, so 12 1 12 2 12 1 2 2 2 F − F � X(x) + Y(y) for some X, Y ∈ C(R ). Since that there exists a F ∈ BC(R ) and Fn ⟶ F in ‖·‖∞. Let 1 2 2 (F − F ) ∈ B (R ), for a fixed y 2 1 2 C 0 f � z12F. &en, f ∈ AC(R ) and ‖fn − f‖ � ‖Fn − F‖ f f ( 2) ‖·‖ 0 � lim F1 − F2 � � lim X(x) + lim Y y0 �. (3) ∞ ⟶ 0. &is shows n ⟶ ∈ AC R in and x⟶− ∞ x⟶− ∞ x⟶− ∞ hence the lemma. □

&erefore, limx⟶− ∞X(x) � − Y(y0) for any x ∈ R. &is X(x) x Alexiewicz norm dose not induce an inner product in implies that is a constant function. For fixed 0, letting 2 y − Y(y) AC(R ) because the norm does not satisfy the parallelogram ⟶ ∞ gives is a constant function for the same 2 law. &erefore, AC(R ) is not a Hilbert space under the norm argument. &ose constant functions sums into 0, and hence 2 F � F ‖·‖. However, the space AC(R ) with ‖·‖ is isomorphic to 1 2. □ 2 BC(R ) with ‖·‖∞. &e mapping f ⟶ F gives a linear &e uppercase letter is denoted the primitive of any isomorphism between the spaces. Uniqueness of primitive 2 integrable distribution f. If f ∈ AC(R ) with the primitive gives the mapping is injective, and by the definition it is F ( 2) ( 2) surjective mapping. Furthermore, the mapping preserves norm ∈ BC R , then for any test function ϕ ∈ D R , 2 ∞ ∞ since ‖f‖ � ‖F‖∞. &us, the space AC(R ) is identified with 〈f, ϕ〉 �〈z12F, ϕ〉 �〈F, z12ϕ〉 � � � Fz12ϕ, (4) the space of continuous functions vanish at infinity. − ∞ − ∞ 2 where it uses HK-integral, and here onward the same, unless Proposition 1. (AC(R ), ‖·‖) is isometrically isomorphic to 2 (B (R2), ‖·‖ ) otherwise stated. Since the linearity of derivatives, AC(R ) C ∞ . forms a vector space with usual operations of functionals. To 2 Next, we define the integral of an integrable distribution, get Banach space structure of AC(R ), it requires to define 2 2 the suitable norm. Using the unique primitive of as it uses the primitive F ∈ BC(R ) of f ∈ AC(R ). f A (R2) ∈ C , we define the Alexiewicz norm as 2 � x y � Definition 2 (see [15], Definition 4.3) Let f ∈ AC(R ) with � � 2 2 ‖f‖ � sup��� � f�: (x, y) ∈ R � � sup |F| �‖F‖ . the primitive F ∈ BC(R ). We define integral of f on I � � � ∞ 2 − ∞ − ∞ R2 [a, b] × [c, d] ⊆ R by (5) b d � f � � � f � F(a, c) + F(b, d) − F(a, d) − F(b, c). Note that the space HK with the Alexiewicz norm is a I a c normed space, but not a Banach space [12, 22]. Absence of (8) Banach space structure of HK is main disadvantage on it. It 2 is one of the reasons to move into AC(R ). &is integral is uniquely defined by uniqueness of the 2 primitive, &eorem 1. Also, it is very general and includes Theorem 2. A (R ) is a Banach space with the Alexiewicz C the Riemann integral, Lebesgue integral, and HK-integral. norm ‖·‖. Norm preservation of Proposition 1 gives that the corre- sponding integral values agree on each integral as presented Proof. First, prove ‖·‖ is a norm. in &eorem 3. 4 International Journal of Mathematics and Mathematical Sciences

x y D(R2) f L1(R2) f A (R2) F(x, y) � � � f Theorem 3. For any test function ϕ ∈ , if ∈ (ii) Let ∈ C and − ∞ − ∞ . 8en is identity with the corresponding distribution 2 F ∈ BC(R ) and f � zF f: ϕ ⟼ (L)�R2 fϕ (or with HK-integral), then 2 (i) f ∈ AC(R ) Proof. See [15]. □ 2 1 2 2 (ii) AC(R ) is the completion of L (R ) (or HK(R )) with respect to the norm: 3. Multiplier for Integrable Distributions � x y � � � 2 ‖f‖ � sup��(L) � � f�: (x, y) ∈ R �. (9) We will turn now to our discussion on multipliers for � − ∞ − ∞ � 2 AC(R ). A multiplier for a class of functions F is a function g such that fg ∈ F for each f ∈ F. For an example, in (or in the sense of HK-integral) Lebesgue integral, g ∈ L∞(Rn) if and only if fg ∈ L1(Rn) � f � (L)� f(or � (HK)� f) 1 n ∞ n (iii) R2 R2 R2 for each f ∈ L (R ). i.e., the space L (R ) is multipliers for L1(Rn). For HK-integral in one-dimensional case 2, space of BV HK Proof bounded variation functions, , is multipliers for [9], (&eorem 6.1.5). In multidimension case, there is no 2 (i) Let f ∈ L1(R2). For any (x, y) ∈ R , we define unique way to extend the notion of variation to function. x y Adams and Clarkson [23] give six such extensions but FL(x, y) � (L) � � f. &en, FL is a primitive − ∞ − ∞ mostly recognized by Vitali variation and Hardy–Krause of f in the Lebesgue integral (or HK-integral). variation. To establish integration by parts formula and &en, FL ∈ C0(R2) since the primitive of Lebesgue 2 multipliers for AC(R ), we begin with Hardy–Krause integral (or HK-integral) is continuous. Clearly, variation. 2 2 FL(− ∞, y) � FL(x, − ∞) � 0 for any (x, y) ∈ R . Two intervals in R are nonoverlapping if their inter- 2 2 section is of Lebesgue measure zero. A division of R is a &erefore, f ∈ AC(R ). 1 2 2 2 finite collection of nonoverlapping intervals whose union is (ii) From (i), L (R ) ⊂ A (R ). Let any f ∈ A (R ). 2 2 C C R . If φ: R ⟶ R, then its total variation (in the sense of 1 2 &en, it needs to show f ∈ L (R ). We prove this by Vitali) is given by 1 2 showing f is a limit point of L (R ). Since � � 2 2 � � f ∈ AC(R ) there is F ∈ BC(R ). Now choose a V12φ � sup ��φai, ci � + φbi, di � − φai, di � − φbi, ci ��, D 2 2 i sequence �Fn � ⊂ C (R ) so that ‖Fn − F‖ ⟶ 0. ∞ (10) Since Fn converges to F uniformly and F(− ∞, y) � F(x, − ∞) � 0, we can assume that D R2 Fn(− ∞, y) � Fn(x, − ∞) � 0 for each n. Now, take where the supremum is taken over all divisions of and I � [a , b ] × [c , d ] D fn � zFn in the distributional sense. &en, interval i i i i i ∈ . 2 1 2 fn � zFn ∈ C(R ) ⊂ L (R ). Moreover, ‖fn − f‖ � Definition 3 (Hardy–Krause bounded variation). A function ‖Fn − F‖∞ ⟶ 0. Hence, f is a limit point of 2 φ: R ⟶ R is said to be Hardy–Krause bounded variation L1(R2). if the following conditions are satisfied. (iii) Any Lebesgue integrable function is HK-integrable with the same value. &us, the result. □ (i) V12φ is finite

(ii) &e function x ⟼ φ(x, y0) is bounded variation on Note that if f ∈ L1(R2), then its Alexiewicz norm and L1 R for some y0 ∈ R norm are not equivalent, for an example see [15] (pg. 9). &e y (x , y) norms are equivalent if f ≥ 0 for almost everywhere. From (iii) &e function ⟼ φ 0 is bounded variation 2 on R for some x ∈ R the definition of integral of f ∈ AC(R ) and its unique 0 primitive, we obtain following fundamental theorem of &e set of functions in Hardy–Krause variation is calculus. 2 2 denoted by BVHK(R ). As in [15], BVHK(R ) forms a Banach space with norm defined by Theorem 4 (fundamental theorem of calculus). ‖φ‖ � ‖φ‖ + ‖V φ‖ + ‖V φ‖ + V φ. Here, V φ and x y BV ∞ 1 ∞ 2 ∞ 12 1 F C(R2) � � zF � F(x, y) V (i) If ∈ then − ∞ − ∞ 2φ are bounded variation to the functions of (ii) and (iii) in + F(∞, ∞) − F(− ∞, y) − F(x, − ∞) Definition 3. For any multidimensional case, any function of International Journal of Mathematics and Mathematical Sciences 5

2 Hardy–Krause variation is a multiplier for HK-integrable Definition 4 (integration by parts). Let f ∈ AC(R ) with its 2 2 functions, see [9]. &erefore, define following integration by primitive F ∈ BC(R ). If g ∈ BVHK(R ), then define 2 parts, as a two-dimensional case which is presented in ([9], integration by parts on [a, b] × [c, d] ⊆ R . &eorem 6.5.9).

b d � � fg � F(a, c)g(a, c) + F(b, d)g(b, d) − F(a, d)g(a, d) − F(b, c)g(b, c) a c b b − � F(x, d)d1g(x, d) + � F(x, c)d1g(x, c) a a (11) d d − � F(b, y)d2g(b, y) + � F(a, y)d2g(a, y) c c b d + � � F(x, y)d12g(x, y). a c

〈τ(s,t)f, ϕ〉 �〈f, τ(− s,− t)ϕ〉 Here, dg(x, y) is Henstock–Stieltjes integral of relevant variables, see in detail [24]. &is integration by parts formula � � � f(x, y)ϕ(x + s, y + t)dxdy induces the space of Hardy–Krause bounded variation R R BV (R2) A (R2) functions, HK , as a multiplier for C . &is � � � f(x − s, y − t)ϕ(x, y)dxdy multiplier is important to define convolution and obtain R R convergence theorem. � � � z12F(x − s, y − t)ϕ(x, y)dxdy It needs to obtain continuity of the Alexiewicz norm R R since the Dirichlet boundary condition is taken on it. For any 2 2 2 (s, t) ∈ R , the translation of f ∈ AC(R ) ⊆ D (R ) is de- � � � z12�τ(s,t)F(x, y) �ϕ(x, y)dxdy. ′ R R fined by 〈τ f, ϕ〉 � 〈f, τ ϕ〉 for ϕ ∈ D(R2), where (s,t) (− s,− t) (12) τ(s,t)ϕ(x, y) � ϕ(x − s, y − t). Translation is invariant and continuous under the Alexiewicz norm. 2 &erefore, τ(s,t)F ∈ BC(R ) is the primitive of 2 2 τ(s,t)f. Theorem 5. Let f ∈ AC(R ) and (s, t) ∈ R . 8en, (i) 2 (ii) Clearly, ‖F‖ � ‖τ F‖ for any (s, t) ∈ R2. τ(s,t)f ∈ AC(R ), (ii) ‖f‖ � ‖τ(s,t)f‖, and (iii) ∞ (s,t) ∞ &erefore, ‖f‖ � ‖τ f‖. ‖f − τ(s,t)f‖ ⟶ 0 as (s, t) ⟶ (0, 0). i.e., continuity of f in (s,t) 2 2 the Alexiewicz norm. (iii) If f ∈ AC(R ), then (f − τ(s,t)f) ∈ AC(R ) so that

2 2 Proof. Let F ∈ BC(R ) be the primitive of f ∈ AC(R ). &en, τ(s,t)F is a continuous function. (i) For any ϕ ∈ D(R2), applying change of variable in integral,

� � � � �f − τ f� ��F − τ F� � sup |F(x, y) − F(x − s, y − t)| ⟶ 0, (s,t) (s,t) ∞ (13) (x,y)∈R2

2 when (s, t) ⟶ (0, 0) because of uniform continuity of F. □ then τ(− s,− t)g ∈ BVHK(R ). From integration by parts, the convolution

f ∗ g(x, y) � � � fτ(− s,− t)gdsdt 4. Integrable Distributional Solution for R R Poisson Equation (14) � � � f(x − s, y − t)g(x, y)dsdt Poisson’s kernel and its convolution are used to find a R R 2 2 classical solution to the Dirichlet problem. In this section, we exists for f ∈ A (R ) and g ∈ BV (R ). Proposition 2 2 C HK use convolution and convergence theorem on AC(R ) to gives some basic properties of convolution which help with 2 find integrable distributional solution. If g ∈ BVHK(R ), our problem. 6 International Journal of Mathematics and Mathematical Sciences

2 2 2 Proposition 2. If f ∈ AC(R ) and g ∈ BVHK(R ), then Theorem 6. Let g ∈ AC(R ) and define u(x, y, z) by (18). (i) f ∗ g exists on R2 and (ii) f ∗ g � g ∗ f. 8en, (i) For each x, y ∈ R and z > 0, the convolution Proof. See [15], &eorem 14.1. □ K(x, y, .) ∗ g(x, y) � u(x, y, z) exists and 2 u ∈ AC(R ) Convergence theorems on Banach space are important 3 (ii) △u(x, y, z) � 0 in R+ when solving partial differential equations. &ere are several (iii) lim ‖u(x, y, z) − g(x, y)‖ � 0, for x, y ∈ R convergence theorems in AC such as strong convergence in z⟶0+ AC, weak/strong convergence in BV, and weak/strong convergence in D. Details are discussed in the plane by Ang, Proof Schmitt and Vy [14] and on R by Talvila [2]. Now, we move 2 into the convergence theorem on A (R ) which will help to C x, y R z interchange limit of integral. (i) Fix ∈ and > 0. &en, the function (s, t) ⟼ K(x − s, y − t, z) is of Hardy–Krause variation on R2. From Proposition 2, K ∗ g exists 2 Proposition 3. Let φn be a sequence in BVHK(R ) so that on the upper half space. Also, u � K ∗ 2 2 ‖φn‖BV < ∞ and limn φn � φ pointwise on R for a g ∈ A (R ) because any function of Hardy–K- ⟶∞2 2 C function φ. If g ∈ AC(R ), then g ∈ BVHK(R ) and rause variation is a multiplier for HK-integrable functions. lim � � gφn � � � gφ. (15) n⟶∞ R R R R (ii) Fix x, y ∈ R and z > 0. To get ux, let xn ⟶ x 2 be a sequence so that xn ≠ x. For any (s, t) ∈ R , define Proof. See [15], Proposition 9.1. □ K xn − s, y − t, z� − K(x − s, y − t, z) φn(s, t) � . Let us move into the main problem. As in [25] (pg. 84), xn − x Poisson’s equation with Dirichlet boundary condition is (19) easily reduced to the case where either f � 0 or g � 0 in (1). &erefore, consider integrable distributional solution for the &en, ux � lim � � g(s, t)φn(s, t)dsdt. Since following problem: n R R K(x − s, y − t, z) of Hardy–Krause bounded vari- 3 2 ⎧⎨ △u � 0, in R+, ation, so does φn(s, t) for each n. Since g ∈ AC(R ), (16) ⎩ 3 from Proposition 3 u � g, on zR+, R3 �� (x, y, z) R3: z � z where + ∈ > 0 . For > 0, consider ux � � � lim g(s, t)φn(s, t)dsdt R R n⟶∞ Poisson’s kernel for the upper half space in 3-dimension (20) − (3/2) z� x2 + y2 + z2 � � � � g(s, t)Kx(x − s, y − t, z)dsdt. K(x, y, z) � . (17) R R 2π Repeat the same argument to obtain uxx, uy, uyy, uz, Note that � � Kdxdy � 1. If g ∈ C(R2) ∩ L∞(R2), R R and uzz. From the linearity of integral, then classical solution for the problem is given by the fol- lowing convolution: △u(x, y, z) � � � g(s, t)△K(x − s, y − t, z)dsdt. u(x, y, z) � K(x, y, .) ∗ g(x, y) R R (18) (21) � � � K(x − s, y − t, .)g(s, t)dtds. R R But Poisson kernel is harmonic, so that 3 &is convolution gives integrable distributional solution △u(x, y, z) � 0 in R+. for (16). International Journal of Mathematics and Mathematical Sciences 7

(iii) Let any α, β ∈ R. For Poisson kernel �R�RKdxdy � 1. Using Fubini–Tonelli theorem and interchanging iterated integral,

α β α β � � (u − g)dxdy � � � [u(x, y, z) − g(x, y)]dxdy − ∞ − ∞ − ∞ − ∞

α β � � � [g(x, y) ∗ K(x, y, z) − g(x, y)]dxdy − ∞ − ∞

α β � � � � � K(s, t, z)[g(x − s, y − t) − g(x, y)]dsdtdxdy − ∞ − ∞ R R α β z � � � � � [g(x − s, y − t) − g(x, y)] s t x y 3/2 d d d d − ∞ − ∞ R R2π�s2 + t2 + z2 � (22) α β 1 � � � � � [g(x − zs, y − zt) − g(x, y)] s t x y 3/2 d d d d − ∞ − ∞ R R2π�s2 + t2 + 1 �

1 α β � � � � � [g(x − zs, y − zt) − g(x, y)] x y s t 3/2 d d d d R R2π�s2 + t2 + 1 � − ∞ − ∞

≤ � � K(s, t, 1)‖g(x − zs, y − zt) − g(x, y)‖dsdt R R

≤ 2‖g‖.

Since α and β are arbitrary,

‖u(x, y, z) − g(x, y)‖ ≤ � � K(s, t, 1)‖g(x − zs, y − zt) − g(x, y)‖dsdt ≤ 2‖g‖. (23) R R

+ 2 To let z ⟶ 0 , apply dominated convergence theorem &en, |gn(s, t)| ≤ 2‖g‖K(s, t, 1) ∈ HK(R ) since (HK-integral) for the sequence of functions: �R�RK(s, t, 1)dsdt � 1. It gives � � � 1 1 � g (s, t) � K(s, t, 1)�g� x − s, y − t� − g(x, y)�. (24) n � n n �

lim � � gn(s, t)dsdt � lim � K(s, t, 1)‖g(x − zs, y − zt) − g(x, y)‖dsdt n⟶∞ R R n⟶∞ � R

(25) � � � lim gn(s, t)dsdt R R n⟶∞ � � � 1 1 � � � � K(s, t, 1) lim �g� x − s, y − t� − g(x, y)�d s dt. R R n⟶∞� n n �

From part (iii) of &eorem 5, we get limn⟶∞‖g(x − limz⟶0+ ‖u(x, y, z) − g(x, y)‖ � 0, and this completes the (1/n)s, y − (1/n)t) − g(x, y)‖ � 0 so that limit of (23), theorem. □ 8 International Journal of Mathematics and Mathematical Sciences

Next, we give an example which will apply &eorem 6. &en, g is HK-integrable but not absolutely integrable. 2 1 2 Hence, g ∈ AC(R )∖L (R ). From &eorem 6, Example 1. Consider the problem (16) in which sin x sin y ⎪⎧ , x ≠ 0 and y ≠ 0, ⎨⎪ xy g(x, y) � (26) ⎪ ⎩⎪ 0, x � 0 or y � 0.

u(x, y, z) � � � K(x − s, y − t, .)g(s, t)dtds R R (27) z − (3/2) sin s sin t � � � �(x − s)2 +(y − t)2 + z2� � �dtds, R R2π st exists and gives solution for the example. Note that absence of any natural topology. &at is, HK is not a 2 2 Kg ∈ HK(R ) since K ∈ BVHK(R ) is a multiplier for Banach space; nevertheless, there is a natural seminorm HK(R2). However, the solution does not exist with the called Alexiewicz. &is is one of the main reasons to move 2 Lebesgue integral. For contraposition, suppose that into the space of integrable distribution AC(R ), which Kg ∈ L1(R2). &en, it follows from Fubini’s theorem [9] contains HK and leads to a Banach space with Alexiewicz (&eorem 2.5.5) that the function norm. Integrable distribution is defined as derivative (distri- h: s ⟼ K(x − s, y − t, z)g(s, t), (28) 2 butional sense) of function in BC(R ). In this definition, 2 1 1 any f ∈ AC(R ) identifies with the unique primitive belongs to L (R ) for almost all t ∈ R. &at is, 2 F ∈ BC(R ). &is unique primitive allows to define a z sin t sin s 2 h(s) � L1 1 Alexiewicz norm ‖·‖ on AC(R ) and leads to a Banach � � 3/2 ∈ �R � 2πt s�(x − s)2 +(y − t)2 + z2� space. Alexiewicz norm does not induce an inner product in 2 AC(R ) because it does not satisfy the parallelogram law. sin s 2 � h (s)� �, &erefore, AC(R ) is not a Hilbert space under the norm 0 s 2 ‖·‖. However, the space (AC(R ), ‖·‖) is isometrically iso- (29) 2 2 morphic to the space (BC(R ), ‖·‖∞). Integral on AC(R ) h (s) � (z t t)( [(x − s)2 + (y − t)2 + z2]3/2) 2 where 0 sin /2π 1/ . is uniquely defined by uniqueness of the primitive. AC(R ) 1 1 &is contradicts that h(s) ∉ L (R ). To see that, suppose is the completion of L1(R2) (or HK(R2)) with respect to 1 1 1 ∞ h(s) ∈ L [0, 1] ⊂ L (R ). Since h0(s) ∈ L [0, 1] is multi- the corresponding Alexiewicz norm. 1 1 plier for L [0, 1], it follows that sin s/s ∈ L [0, 1]. &is im- In one-dimensional HK-integral, space of bounded plies that sin s/s absolutely integrable on [0, 1], which is a variation functions, BV, is multipliers for HK. In mul- 1 1 contradiction. Hence, h(s) ∉ L (R ). tidimension case, there are several extensions to variation of function. To establish integration by parts formula and Even HK-integral exists in (27), it may not be able to 2 multipliers for AC(R ), we begin with Hardy–Krause evaluate and get explicit function. In that case, numerical variation. &e space of Hardy–Krause bounded variation integration can be applied. Numerical integration should be 2 2 functions, BVHK(R ), as a multiplier for AC(R ). &is done with respect to s and t while fixing x, y, and z, details in multiplier is important to define convolution and obtain [26]. &is reduces solving the partial differential equation 2 convergence theorem. BVHK(R ) is used to define con- into applying numerical integration for the double integral 2 volution and obtain convergence theorem on AC(R ). (27). &ere are several convergence theorems in AC such as strong convergence in AC, weak/strong convergence in BV, and weak/strong convergence in D. Convergence 5. Discussion 2 theorem on AC(R ) and dominated convergence theorem In this work, integrable distributional solution is obtained in HK(R2) are used to obtain integrable distributional for Poisson equation with the Dirichlet boundary condition solution. Finding of this article gives solution for Poisson in the upper half space. Boundary conditions are taken in the equation for broader initial condition in the upper half Alexiewicz norm. Lebesgue integral uses for the classical space. Dirichlet problem in R3 is considered, and the so- solution to Poisson equation. Because of the proper inclu- lution has obtained by HK-integration where it is not sion L1 ⊊ HR, we may use HK-integral to solve Poisson possible from Lebesgue integral. Because of function’s equation. However, there are some limitations due to the complexity, it may not be able to have solution in explicit International Journal of Mathematics and Mathematical Sciences 9 form. It would help keep an analytical outlook in the so- [13] P. Mikusinksi´ and K. Ostaszewski, “Embedding Henstock lution, which would be better than solving the partial dif- integrable functions into the space of Schwartz distributions,” ferential equation by numerical methods such as finite Real Analysis Exchange, vol. 14, no. -89, pp. 24–29, 1988. difference methods and finite element methods. [14] D. D. Ang, K. Schmitt, and L. K. Vy, “A multidimensional &ere is potential to find integrable distributional so- analogue of the denjoy–perron–henstock–kurzweil integral,” 8e Bulletin of the Belgian Mathematical Society — Simon lution for Poisson equation in other boundary conditions, Stevin, vol. 4, pp. 355–371, 1997. Neumann and Robin. But it needs to develop suitable [15] E. Talvila, “&e continuous primitive integral in the plane,” convergence theorems on AC and its multipliers BVHK. Real Analysis Exchange, vol. 45, no. 2, pp. 283–326, 2020. Also, it is possible to consider Poisson equation in unit ball [16] S. Heikkil¨a, “On nonlinear distributional and impulsive with the relevant Poisson kernel. Talvila, in [15], gives some Cauchy problems,” Nonlinear Analysis: 8eory, Methods & construction to develop integrable distribution on Rn, in- Applications, vol. 75, no. 2, pp. 852–870, 2012. tegration by parts, and mean value theorem and defines [17] B. Liang, G. Ye, W. Liu, and H. Wang, “On second order n n AC(R ) and BC(R ) for multidimensional case. Hence, nonlinear boundary value problems and the distributional integrable distributional solution is possible to obtain for Henstock-Kurzweil integral,” Boundary Value Problem, n vol. 2015, 2015. Poisson equation with Dirichlet boundary condition in R+ with suitable convergence theorem. [18] A. Sikorska-Nowak, “Existence theory for integrodifferential equations and henstock-kurzweil integral in Banach spaces,” Journal of Applied Mathematics, vol. 2007, Article ID 31572, Data Availability 12 pages, 2007. [19] M. 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