The Dirichlet problem for certain discrete structures

The Dirichlet problem for certain discrete structures

Abtin Daghighi

U.U.D.M. Project Report 2005:4

Examensarbete i matematik, 20 poäng Handledare och examinator: Christer Kiselman

April 2005

Department of Mathematics

The Dirichlet problem for certain discrete structures

Abtin Daghighi

Abstract

The paper reviews some of the early ideas behind the development of the

theory of discrete harmonic functions. The connection to random walks is

pointed out. Then a current and more general construction using weight

functions is described. Discrete analogues of the Laplace operator are

defined for Z ⁿ and a discrete planar hexagonal structure H. Discrete ana-

logues of the Dirichlet problem and Poisson’s equation are formulated and

existence and uniqueness of bounded solutions is proved for the finite case

and also for a certain case of infinite sets in Z ⁿ . Explicit discrete analogues

of the Green function are presented for Z ³ and Z[i] equipped with a certain

discrete calculus, which is also briefly presented.

Contents

1 Some results from the continuous case 3

1.1 The Dirichlet principle . . . . 3 1.2 The Laplace equation and the maximum principle . . . . 4 1.3 The method of Perron . . . . 6

2 Introduction to the discrete case 7

3 The Dirichlet problem for subsets of Z ⁿ 8

3.1 Separation of variables . . . . 12

4 Connection to martingales in two dimensions 14

5 A discrete analogue of Poisson’s equation for Z ³ 14

6 A discrete calculus for Z[i] 17

6.1 A discrete Green function for Z[i] . . . . 20 7 A more general Laplace operator and an analogue to the method

of Perron 24

8 A hexagonal planar structure 27

9 References 28

10 Notation 29

1 Some results from the continuous case

In this section some of the theory of partial differential equations which is used in connection with the Dirichlet problem will be presented. In some cases the ideas behind these results have been used to formulate and prove discrete analogues for functions defined on discrete structures, which is the main topic of this paper.

Therefore it might be useful to first review some of the theory of this section to better understand the background of the chapters that follow.

1.1 The Dirichlet principle

The Dirichlet problem for Laplace’s equation on Ω ⊂ R ⁿ consists of finding a function u which satisfies the following conditions

(∗)

( ∆u(x) = 0, x ∈ Ω;

u(x) = g(x), x ∈ ∂Ω.

The inhomogeneous Laplace equation is known as Poisson’s equation. The Dirichlet problem for Poisson’s equation is to find a function u ∈ C ² (Ω), such that for a given function, f ∈ C ⁰ (Ω), the following conditions hold

(∗∗)

( ∆u(x) = f (x), x ∈ Ω;

u(x) = 0, x ∈ ∂Ω.

Note that by adding a solution of the Dirichlet problem for the Laplace equa- tion on the same set one gets a solution for an inhomogeneous continuous bound- ary condition.

The problem (∗) can be considered as a minimization problem according to the following principle.

The Dirichlet principle. Let

S = {w ∈ C ² (Ω), w = g on ∂Ω}

and let

I(w) = 1 2

Z

Ω

|∇w| ² dx.

A function u ∈ S is a solution to the Dirichlet problem for Laplace’s equation on Ω, (∗), if and only if

I(u) = min

w∈S I(w).

Proof. Suppose u is a solution of (*). Then for any w ∈ S, 0 =

Z

Ω

∆u(u − w)dx = − Z

Ω

|∇u| ² dx + Z

Ω

∇u∇wdx ≤

− Z

Ω

|∇u| ² dx + 1 2

Z

Ω

|∇u| ² + |∇w| ²
dx =

− 1 2

Z

Ω

|∇u| ² dx + 1 2

Z

Ω

|∇w| ² dx.

Therefore

Z

Ω

|∇u| ² dx ≤ Z

Ω

|∇w| ² dx, and since w ∈ S was arbitrary

I(u) = min

w∈S I(w).

Next suppose that w ∈ S minimizes I. Let v ∈ C ² be such that v ≡ 0 for x ∈ ∂Ω.

Then u + v ∈ S for all . Define

J () ≡ I(u + v).

Since u minimizes I, J must have minimun for  = 0. Now J () =

Z

Ω

|∇u| ² + 2∇u∇v +  ² |∇v| ²
dx, implies

J ⁰ () = Z

Ω

∇u∇v + 2|∇v| ² dx, so that

J ⁰ (0) = − Z

Ω

(∆u)vdx + Z

∂Ω

∂u

∂ν vdσ(x) = − Z

Ω

(∆u)vdx Now J ⁰ (0) = 0 implies

0 = Z

Ω

(∆u)vdx,

for all v ∈ C ² (Ω) such that v = 0 for x ∈ ∂Ω, thus ∆u = 0. This completes the proof.

1.2 The Laplace equation and the maximum principle

What follows is the derivation of a fundamental solution for the Laplace equation.

Make the ansatz of a radial symmetric (around the point ξ ∈ R ⁿ ) harmonic function, v(r) ∈ C ² (Ω), r = |x − ξ|, as a solution for (∗). The chain rule gives

∆v(r) = v ⁰⁰ (r) + n − 1

r v ⁰ (r) = 0,

which implies



 

 

v ⁰ (r) = Ar ¹⁻ⁿ

v(r) = ^Ar _2−n

, n 6= 2 v(r) = A ln r, n = 2

Introducing the ball, B(ξ, ρ) ⊂ Ω, with surface, S(ξ, ρ), and using the Green identity

Z

Ω

(v∆u − u∆v) dx = Z

∂Ω

 v ∂u

∂n − u ∂v

∂n

 dx, gives

Z

Ω

(v∆u − u∆v) dx = Z

∂Ω

 v ∂u

∂n − u ∂v

∂n

 dS +

Z

S(ξ,ρ)

 v ∂u

∂n − u ∂v

∂n

 dS = Z

∂Ω

 v ∂u

∂n − u ∂v

∂n



dS − v(ρ) Z

B(ξ,ρ)

∆udx − v ⁰ (ρ) Z

S(ξ,ρ)

udS

→ Z

∂Ω

 v ∂u

∂n − u ∂v

∂n



dS + Aω _n u(ξ), where in the second equality the following is used: ^dv(ρ) _dn = ˆ n · v ⁰ (r) = −v ⁰ (ρ), and where ω n = _Γ(n/2) ^2π

, i.e., the surface area of the unit sphere in R ⁿ , A ∈ R. So by choosing the constant A ∈ R to be _ω ¹

n

one gets u(ξ) = −

Z

∂Ω



v(x, ξ) ∂u(x)

∂n − u(x) ∂v

∂n

 dS +

Z

Ω

v(x, ξ)∆u(x)dx.

In particular, replacing u(x) by φ(x) ∈ C ₀ ^∞ (Ω) shows that ∆v(x, ξ) = δ(ξ), in the distribution sense, which makes it a fundamental solution of the Laplace operator.

Further if there exists a fundamental solution say, w(x, ξ), such that w(x, ξ) = 0 for x ∈ ∂Ω, ξ ∈ Ω, then replacing v by w in the last equation yields a solution to (∗) and also to (∗∗), namely

u(ξ) = Z

∂Ω

g(x) ∂w

∂n (x, ξ)dS + Z

Ω

w(x, ξ)f (x)dx.

A function f satisfying ∆f (x) ≥ 0 for all x ∈ Ω is called subharmonic on Ω.

We shall now show a version of a result known as the maximum principle for subharmonic functions. This will be done in two steps.

In the first step consider a function g ∈ C ² (Ω) ∩ C ⁰ (Ω) satisfying ∆g > 0 in Ω. If g assumes a maximum at ξ ∈ Ω ^◦ then _∂x ^∂

^g

k

≤ 0 at ξ for all k , which implies

∆g ≤ 0. This is a contradiction to the assumption ∆g > 0 in Ω. Thus max

Ω

g = max

∂Ω g.

The next step is to consider u ∈ C ² (Ω) ∩ C ⁰ (Ω) satisfying ∆u ≥ 0 in Ω. Let

 > 0, and g = |x| ² . Then ∆g > 0, and so the result of the first step applies to

u + g. Hence one gets

max

Ω

(u + g) = max

∂Ω (u + g), which implies

max

Ω

u ≤ max

Ω

(u + g) = max

∂Ω

(u + g) ≤ max

∂Ω u +  max

∂Ω g.

Letting  → 0 gives

max

Ω

u ≤ max

∂Ω u.

This is the required result.

1.3 The method of Perron

The method of Perron for findning solutions for (∗) uses the space of subharmonic functions over Ω, which is denoted by

SH(Ω) = {u ∈ C ² (Ω); ∆u(x) ≥ 0, x ∈ Ω}.

Now for a given g ∈ C ⁰ (∂Ω), define

SH _g (Ω) = {u ∈ C ⁰ (Ω) ∩ SH(Ω); u(x) ≤ g(x), x ∈ ∂Ω}.

Further define w _g = sup _u∈SH

_(Ω) u(x). Since the results from the continuous case are only given to provide a preliminary background and are not the main topic of this paper, only a brief summary of the method of Perron will be given. Several statements will thus be given without proof. Define

( u B (x) = u(x), x ∈ Ω, x / ∈ B(ξ, r),

∆u B (x) = 0, x ∈ B(ξ, r),

where u ∈ C ⁰ (Ω), B(ξ, r) ⊂ Ω, r > 0, ξ ∈ Ω. Such a u _B will have the following property

u(x) ≤ u _B (x), x ∈ Ω, u _B ∈ SH(Ω).

The existence and uniqueness of such a u _B is stated here without proof; cf. John (1982). The next step is to show that w _g is harmonic in Ω. To do this, start by constructing a sequence of functions that are harmonic in B(ξ, r) and lie between fixed bounds. Let x ^k , k = 1, 2, . . . , be a sequence in B(ξ, r ⁰ ) ⊂ Ω , r ⁰ < r. Then there exists functions, u ^j _k ∈ SH _g (Ω), such that lim _j→∞ u ^j _k (x ^k ) = w _g (x ^k ), since w _g is defined as the supremum. Define u ^j (x) = max(u ^j ₁ (x), . . . , u ^j _j (x)) ∈ SH g (Ω).

Now u ^j (x) ≥ u ^j _k (x) for x ∈ Ω, j ≥ k. Then also

(∗ ∗ ∗) lim _j→∞ u ^j (x ^k ) = w _g (x ^k ), for all k.

Replacing if necessary u ^j by max(inf g, u ^j ) provides a sequence u ^j (x) such

that inf g(x) ≤ u ^j (x) ≤ sup g(x), x ∈ Ω. Then replacing u ^j by u ^j _B , guarantees

that each function of the series is harmonic in B(ξ, r), and the limit is the same

as the original sequence. The bounds form a compact domain. Thus the wanted sequence is now constructed. Then it is stated here without proof that (cf. for example John 1982) there exists a convergent subsequence with a harmonic limit function, W , for x ∈ B(ξ, r ⁰ ) ⊂ B(ξ, r). (∗ ∗ ∗) then implies that w g (x ^k ) = W _g (x ^k ) for all k = 1, 2, . . . The next step is to show that w _g agrees with W in a neighborhood of ξ. Let first x ^k , k = 1, 2, . . . be a sequence converging to x ∈ B(ξ, r ⁰ ). Continuity of W implies that lim k→∞ w g (x ^k ) exists for all k = 1, 2, . . . so w _g (x ^k ) is continuous in B(ξ, r ⁰ ). Then taking for x ^k a sequence which is dense in B(ξ, r ⁰ ) proves that w _g = W in B(ξ, r ⁰ ) since both functions are continuous on B(ξ, r ⁰ ). Hence w g is harmonic in a neighborhood of ξ, which was arbitrary. This implies that w _g is harmonic in Ω.

In order to proceed, the concept of a barrier must first be defined. For y ∈ ∂Ω a barrier function, q y (x), is a function in C ⁰ (Ω) ∩ SH(Ω), such that

( q _y (y) = 0,

q _y (z) < 0, z ∈ ∂Ω, z 6= y.

The so called barrier postulate states that there exists a barrier function for every y ∈ ∂Ω. The existence of a barrier is in fact a condition that ensures that the constructed w _g actually is a solution to (∗). This will be stated in the form of a theorem in this paper without proof; cf. for example John (1982).

Theorem. Let there exist for each y ∈ ∂Ω, a function q _y ∈ C ⁰ (Ω) ∩ SH(Ω), such

that (

q _y (y) = 0,

q _y (z) < 0, z ∈ ∂Ω, z 6= y.

Then lim x∈Ω,x→y w g (x) = f (y).

Defining w _g (x) = g(y), y ∈ ∂Ω, implies w _g ∈ C ⁰ (Ω) and hence provides a solution for (∗).

2 Introduction to the discrete case

The purpose of this paper is to review definitions for describing discrete analogues of the Dirichlet problem and Poisson’s equation for certain discrete structures, and also to review some of the results of the theory.

The theory of difference equations is important in many applications such as optimal system control and signal processing. This is because devices for physical measurements often make registrations at equidistant discrete time points. Prob- lems involving difference equations also appear in probability theory in connection with stochastic processes, e.g., random walks and in fact the results of McCrae

& Whipple (1940) that are presented in this paper have developed through the

study of random walks. It is also pointed out by Phillips & Wiener (1923) that

a certain type of problem concerning the electric potential at discrete points of

an electric circuit, which are connected by wires of equal resistance, may be dis-

cribed in terms of a difference operator. This is also the reason that the theory

which was laid forth concerning this problem was called potential field theory. The ideas of Wiener where further developed by Ferrand (1944) and Duffin (1952), who adapted the theory to make it easier to examine further discrete analogues of operators from the theory of partial differential equations and complex analysis.

The focus of this paper will be on sets isomorphic to Z ⁿ and with a fixed distance between two adjacent points of the set. These are special cases of constructions involving a parameter that allows variation of the smallest distance and hence allowing investigation of asymptotic behavior of operators defined. The latter constructions play an important role in numerical analysis and are not consid- ered in this paper. Instead, general concepts and ideas that are used to form constructions suitable for developing a theory for discrete harmonic functions are presented. Methods for investigating the existence of solutions to the discrete analogues of the Dirichlet problem and Poisson’s equation for certain cases are considered. Furthermore a sophisticated expansion of the theory by the intro- duction of structural functions shall be presented. These results are given by Kiselman (2005). Here the discrete structure is considered as a graph with ver- tices at each point of a given set, X, and where pairs of points belonging to the edge-set are neighbors. Furthermore a constant, c(x, y), is assigned to each ordered pair of neighbors (x, y), x, y ∈ X. This is done in a not necessarily sym- metric manner, i.e., c(x, y) is not necessarily equal to c(y, x). This means that the Dirichlet problem may in fact be interpreted as a random walk. The relation between the discrete Dirichlet problem and random walks in two dimensions was pointed out by Courant et al. (1923) and also by Kemeny & Snell (1958).

Finally a discrete Laplace operator is defined for a discrete hexagonal planar structure and the Dirichlet problem for Poisson’s equation in this case will be considered. These results come from Rodin (1987).

It should be noted that Chapter 7 generalizes much of the results of the preceding chapters. However these special cases are presented anyway to give a broader understanding of the ideas behind the development of the theory.

3 The Dirichlet problem for subsets of Z ⁿ

In this chapter a short description is given of a structure, called a net, involving a parameter that allows investigation of asymptotic behavior and which is isomor- phic to Z ⁿ . Then the case of a fixed value for the parameter is considered and the construction is then called a lattice. Discrete analogues of the Laplace oper- ator and the Dirichlet problem will then be presented and a theorem concerning existence of solutions to the defined Dirichlet problem will be given. Finally as an example the discrete Dirichlet problem in the case of a finite cube in Z ³ will be solved using the method of separation of variables.

The following is a short review of some definitions and derived relations pre- sented in Phillips & Wiener (1923) and Duffin (1952). A net of order ν over Z ⁿ , where ν ∈ N, consists of the hyperplanes

{x ∈ R ⁿ ; x _j = 2 ^−ν m _j }, m _j ∈ Z, j = 1, . . . , n.

The nodes of the net are points where n such planes intersect. Hence each node has 2n adjacent nodes. A boundary node for the net over a region, R, i.e., the restriction of the net to R, is a node in R such that some adjacent node does not lie in R. An interior node is a node in R that is not a boundary node. The set of interior nodes is denoted by R ^◦ , and the set of boundary nodes is denoted by

∂R.

A harmonic function over a subset Ω of R ⁿ has the mean value property, i.e., its value at a point is the mean value over a sphere:

(3.1) f (x) = Z

kx−yk=r

f (y)dy, x ∈ Ω, {y; kx − yk ≤ r} ⊂ Ω.

An analogue to the condition (3.1) for a function u defined on a net in n dimensions is the following relation

(3.2) 2nu  1

2 ^ν x



=

n

X

i=1

 u  x ₁

2 ^ν , . . . , x _i + 1

2 ^ν , . . . , x _n 2 ^ν



+ u  x ₁

2 ^ν , . . . , x _i − 1

2 ^ν , . . . , x _n 2 ^ν



Phillips & Wiener (1923) call a function satisfying (3.2) a potential function on the net. The construction of a net is useful when examining the behavior of any defined operator to its continuous analogue, hence letting ν → ∞. Duffin (1952) denotes by a lattice the space Z ⁿ with lattice points at points whose coordinates are integers (hence ν = 0 when comparing to a net).

Henceforth the term grid will be used in this paper instead of lattice or net for the same structure. The terms node and point have equivalent meaning for any grid. Any subset of a grid will again be called a grid. In the case of ν = 0, condition (3.2) can be expressed in terms of a difference operator, ∆ _n , as follows (3.3)

∆ _n u =

n

X

i=1

{u(x ₁ , . . . , x _i + 1, . . . , x _n ) + u(x ₁ , . . . , x _i − 1, . . . , x _n )} − 2nu(x) = 0.

A function satisfying (3.3) for each point on a subset of a lattice is called discrete harmonic on that subset. Let Ω ⊂ Z ⁿ be a grid. A discrete analogue of the Dirichlet problem for Ω is to find a function u on the grid satisfying the following relations

∆ _n u(x) = 0, x ∈ Ω ^◦ , (3.4)

u(x) = g(x), x ∈ ∂Ω.

(3.5)

Assume that a function, u, is discrete harmonic on a grid Ω ⊂ Z, and let A be

the maximum value attained by u(x), x ∈ Ω. If u(y) = A, y ∈ Ω ^◦ , then f (z) = A

for all z such that z is an adjacent point of y. This follows directly from the fact

that f is discrete harmonic. Hence any discrete harmonic function satisfies the

following principle:

Maximum principle. Suppose that Ω ⊂ Z ⁿ is bounded and that u is harmonic on Ω. Then max _Ω u = max _∂Ω u.

The following theorem shows that this problem has a unique solution in the finite case. In the proof there is a minimization procedure of a quadric form, and this may be considered as an analogue of the Dirichlet principle; cf. section 1.1.

Theorem 1. Let Ω ⊂ Z ⁿ have finitely many interior points and let g be a bounded function on the boundary points. Then the discrete Dirichlet problem (3.4), (3.5) has a unique solution.

Proof. Let Ω ^◦ contain N points and consider the function values of u on the interior points as a finite set of variables. For any solution a system of N equations must be satisfied. Ordering the set of variables in a vector ξ = (u 1 , . . . , u N ) this system can be written in matrix form

M ξ = c

where M is an N × N matrix and where in each equation any boundary function value has been moved to the right hand side, hence c is a constant vector in R ^N .

Now consider the following quadratic form

W = X

n

X

i=1

(u(x ₁ , . . . , x _n ) − u(x ₁ , . . . , x _i − 1, . . . , x _n )) ² ,

where the second summation is over all points of the grid such that the arguments are defined function values, either unknown or known. This means that if x ∈ Ω ^◦ then the square of the difference u(x) − u(y), y any adjacent point to x, will appear exactly once in the sum. But if x ∈ ∂Ω and z is an adjacent point to x which is not in Ω ^◦ then u(x) − u(z) = g(x) − u(z) is either a known constant (if z ∈ ∂Ω) or it does not appear as a term of W (if z / ∈ Ω). Minimizing W with respect to the variables gives

0 = 1 2

∂W

∂u _j =

n

X

i=1

{(u(x) − u(x ₁ , . . . , x _i + 1, . . . , x _n ))+

(u(x) − u(x ₁ , . . . , x _i − 1, . . . , x _n ))}, j = 1, . . . , N.

which is equivalent to (3.2). W ≥ 0 has a finite number of terms and tends to +∞ as kuk → +∞. Hence it must attain a minimum. Now if each component of c is equal to a constant, say A, then W = 0 (which is a minimum since W ≥ 0) iff u(x) = A for all x ∈ Ω. So if c is the zero vector then u = 0 on ∂Ω and W attains the minimum value W = 0 iff u = 0 also on Ω ^◦ , hence ξ = 0. Conversely ξ = 0 implies M ξ = 0 = c. Thus the null space of M consists of the zero vector alone. This means that M is invertible so there exists a unique vector ξ = M ⁻¹ c consisting of the values of u on Ω ^◦ . This completes the proof.

For any grid in Z ⁿ define a path as a set of the following type:

Γ = {(x ₀ , . . . , x _m ); x _i ∈ Ω, i = 0, . . . , m ∈ N, x _i , x _i+1 are adjacent nodes}.

A grid is connected iff any two separate points of the grid are the first and last

points of a path belonging to the grid. In the proofs of the two theorems that follow we shall write x ∼ y whenever y is an adjacent point to x. If a grid has infinitely many interior points one may use a linear operator to represent the infinite set of equations that must be satisfied for a solution of the Dirichlet problem. The following two results are due to the author and deal with an infinite case of the discrete Dirichlet problem.

Theorem 2. Let Ω ⊂ Z ⁿ have infinitely many interior points and let each interior point have at least one adjacent boundary node. Further let g ∈ ` <sup>∞</sup> be a function on the boundary nodes. Then the discrete Dirichlet problem (3.4), (3.5) has a unique bounded solution u ∈ ` ^∞ .

Proof. Since the set of interior points is countable, the values of a function u = u(x), x ∈ Ω, on these nodes may be ordered in the form of a vector υ = (u ₁ , u ₂ , . . .). Then, in each of the infinitely many equations that must be satisfied for a solution, move any boundary function value occurring in the equation (3.3) to the right hand side so that the condition that must be satisfied can be written in the following form

(3.6) Lυ = b

where b ∈ ` ^∞ is a constant vector with components of the form _2n ¹ P

y∈∂Ω y∼x g(y), and L = I − ˜ L, where the components of ˜ Lυ have the form _2n ¹ P

y∈Ω

y∼x

u(y). (3.6) is thus a system of equations where each equation has the following form:

u(x) − 1 2n

X

y∈Ω

y∼x

u(y) = 1 2n

X

y∈∂Ω y∼x

g(y), x ∈ Ω.

Now, by hypothesis, for any x ∈ Ω ^◦ there exists at most 2n − 1 points y ∈ Ω ^◦ such that y is adjacent to x. Using the triangle inequality gives

k ˜ L(u)k ∞ = sup

x

1 2n

X

y∈Ω

y∼x

u(y)    

≤ 2n − 1 2n sup

x∈Ω

|u(x)| =

 1 − 1

2n

 kuk ∞ .

Thus k ˜ Lk < 1 and therefore using the Neumann series one can write (3.7) L ⁻¹ = (I − ˜ L) ⁻¹ = I + ˜ L + ˜ L ² + ˜ L ³ + · · ·

and

(3.8) υ = L ⁻¹ b

is the unique solution of (3.4), (3.5). This completes the proof.

The next theorem shows that the result just given also holds for the case g ∈ ` ¹ .

Theorem 3. Let Ω ⊂ Z ⁿ have infinitely many interior points and let each

interior point have at least one adjacent boundary node. Further let g ∈ ` ¹ be a

function on the boundary nodes. Then the discrete Dirichlet problem (3.4), (3.5)

has a unique solution u ∈ ` ¹ .

Proof. The set of interior points is countable, so the values of a function u = u(x), x ∈ Ω, on these nodes may be ordered in the form of a vector υ = (u ₁ , u ₂ , . . .).

Thus using the same notations as in Theorem 2, but with g ∈ ` ¹ , we have a system of equations that must be satisfied by the wanted solution given by

Lυ = b,

where each equation can be written in the following form u(x) − 1

2n X

y∈Ω

y∼x

u(y) = 1 2n

X

y∈∂Ω y∼x

g(y).

Now, by hypothesis, for any y ∈ Ω ^◦ there exists at most 2n − 1 points x ∈ Ω ^◦ such that y is adjacent to x. This implies that summing over all x ∈ Ω ^◦ when calculating the norm k ˜ Lυk, each such y may appear at most 2n − 1 times in the sum. This implies that

k ˜ L(u)k ₁ = X

x∈Ω

1 2n

X

y∈Ω

y∼x

u(y)    

≤ 2n − 1 2n

X

z∈Ω

|u(z)| =

 1 − 1

2n

 kuk ₁ .

Thus k ˜ Lk < 1 and as in Theorem 2 one can write

L ⁻¹ = (I − ˜ L) ⁻¹ = I + ˜ L + ˜ L ² + ˜ L ³ + · · · , so that

υ = L ⁻¹ b,

is the unique solution of (3.4), (3.5). This completes the proof.

As an example in three dimensions, consider

Ω = {x = (x ₁ , x ₂ , x ₃ ) ∈ Z ³ ; −1 ≤ x _i ≤ 1, i = 1, 2},

and let g be a given bounded function on ∂Ω. Then Theorem 2 implies that the discrete Dirichlet problem has a unique bounded solution for this infinite strip.

3.1 Separation of variables

In this section the discrete Dirichlet problem in the case of a finite cube in Z ³ will be solved using the method of separation of variables. This method is described by Phillips & Wiener (1923).

Consider the discrete Dirichlet problem (3.4), (3.5) for the case n = 3 and a grid consisting of a finite cube with vertices (0, 0, 0), (b, 0, 0), (0, b, 0), (0, 0, b), (b, b, 0), (b, 0, b), (0, b, b), (b, b, b), b ∈ Z ₊ , and with prescribed values on the bound- ary points. The ansatz u(x ₁ , x ₂ , x ₃ ) = X(x ₁ )Y (x ₂ )Z(x ₃ ) gives together with (3.4) the following relation

(3.9) 3 = X( ^x

_b ⁻¹ ) + X( ^x

_b ⁺¹ )

2X( ^x _b

) + Y ( ^x

_b ⁻¹ ) + Y ( ^x

_b ⁺¹ )

2Y ( ^x _b

) + Z( ^x

_b ⁻¹ ) + Z( ^x

_b ⁺¹ )

2Z( ^x _b

) .

It suffices to find a solution which agrees with the given boundary function on one side of the cube and is zero on the remaining five, since the sum of six such functions corresponding to the different sides of the cube is discrete harmonic for all interior points and satisfies the conditions on all boundary points. Let the boundary values on the part of the cube lying in the plane x ₃ = 0 be given by f (x ₁ , x ₂ ). The trigonometric relations

sin kπ( ^x

_b ⁻¹ ) + sin kπ( ^x

_b ⁺¹ )

2 sin kπ ^x _b

= cos kπ x ₁ b sin lπ( ^x

_b ⁻¹ ) + sin lπ( ^x

_b ⁺¹ )

2 sin lπ ^x _b

= cos lπ x ₂ b sinh mπ( ^x

_b ⁻¹ − 1) + sinh mπ( ^x

_b ⁺¹ − 1)

2 sinh mπ( ^x _b

− 1) = cosh kπ( x 3

b − 1), k, l, m ∈ Z, imply that

sin kπx ₁

b sin lπx ₂

b sinh mπ  x ₃ b − 1 

is a solution of (3.4) which vanishes on all boundary points except those lying in the plane x ₃ = 0, provided that

cos kπ x ₁

b + cos lπ x ₂

b + cosh mπ x ₃ b = 3 (3.10)

which shows that m is a function of k, l. Then the linear combination

b−1

X

k=1 b−1

X

l=1

A k,l sin kπx ₁

b sin lπx ₂

b sinh mπ

 x ₃ b − 1 

, (3.11)

also solves (3.4) and vanishes on the surface of the cube except the part in the plane x ₃ = 0. On that plane (3.11) may be considered as the finite Fourier series for f (x ₁ , x ₂ ) giving the relation

(3.12) −A _k,l sinh mπ =

b−1

X

i=1 b−1

X

j=1

f (i, j) sin kπi

b sin lπj b Hence

(3.13) u 1 (x 1 , x 2 , x 3 ) =

b−1

X

k=1 b−1

X

l=1

sin kπx 1

b sin lπx 2

b sinh mπ( x 3

b − 1)×

−

b−1

X

i=1 b−1

X

j=1

f (i, j) sin kπi

b sin lπj b

!

is the part of a complete solution corresponding to one of the six sides of the

cube. The remaining five parts of a complete solution are obtained analogously.

4 Connection to martingales in two dimensions

In this chapter a close connection of the discrete Dirichlet problem in two dimen- sions to that of martingales in the study of random walks shall be disscussed.

The ideas behind this connection was presented by Kemeny & Snell (1958).

A random walk in Z ² is a process which moves from (x ₁ , x ₂ ) ∈ Z ² to (x ₁ +1, x ₂ ), (x 1 − 1, x 2 ), (x 1 , x 2 + 1), (x 1 , x 2 − 1) with equal probabilities, i.e., 1/4. This is a special case of a process where the probabilities of going from state i to state j, is given by p _ij . Thus in the case of a finite number of states the process may be described by the so called transition matrix, P, P ij = p ij . A function on the states can be presented by a column vector u. Such a vector is called a martingale if P u = u. A state, i, is called a boundary state if p _ii = 1. Let B and I be finite sets of grid points such that from any point of I, a random walk reaches a point of B, but cannot reach any point not in B ∪ I without going through B. The B is called a boundary set, and I is called an interior set. Assume that the boundary values f (i, j), are given on B. Then consider the probabilistic problem of finding a grid function, u, defined on B ∪ I, such that u is a martingale, i.e., P u = u. In this special case this explicitly means that u has the properties

(4.1) 4u(i, j) = u(i + 1, j) + u(i − 1, j) + u(i, j + 1) + u(i, j − 1), (i, j) ∈ I, (4.2) u(i, j) = f (i, j), (i, j) ∈ B.

Such a function is in fact the solution to the discrete Dirichlet problem for a finite two-dimensional grid, as it is described in the preceding chapter.

5 A discrete analogue of Poisson’s equation for Z ³

This chapter is concerned with finding a function on a grid, Ω ⊂ Z ⁿ , for the special case of n = 3 such that the function has properties that are analogous to the Green function. Fourier transformations shall be used in the derivations. The chapter presents some of the results found in the reference Duffin (1952). Given a function u = u(a ₁ , a ₂ , a ₃ ) on Ω, a new function w is defined on all interior points by

(5.1) ∆ 3 u = −w,

which is a discrete analogue of Poisson’s equation. Considering the function values as Fourier coefficients gives the formal Fourier series

U (x) = X

a∈Ω

u(a)e ^−ia·x (5.2)

W (x) = X

a∈Ω

w(a)e ^−ia·x

(5.3)

Then the inversion formula for Fourier series gives (5.4) u(a) = (2π) ⁻³

Z Z Z

x

,x

,x

∈[−π,π]

e ^ia·x U (x)dx ₁ dx ₂ dx ₃

The case u(a) = e ^ia·x gives

(5.5) ∆ ₃ u = 6 − e ^ix

− e ^−ix

− e ^ix

− e ^−ix

− e ^ix

− e ^−ix

u(a) = : − P (x)u = 6−2(cos x ₁ +cos x ₂ +cos x ₃ )u(a) = −4 

sin ²  x ₁ 2



+ sin ²  x ₂ 2



+ sin ²  x ₃ 2



u(a)

= u(a)kxk ² + O(kxk ⁴ ), kxk → 0.

In the calculations the following relation has been used: 1 − cos x = 2 sin ^{2 x} ₂
.

Operating with ∆ ₃ on u(a) gives (since the operator acting on u corresponds to multiplication by −P , which means that the formal Fourier series of ∆ ₃ u(a) is P (x)U (x))

(5.6) w(a) = (2π) ⁻³

Z Z Z

x

,x

,x

∈[−π,π]

e ^ia·x P (x)U (x)dx ₁ dx ₂ dx ₃ .

Hence by (5.3) (and since two functions having the same Fourier coefficients are equal)

(5.7) P (x)U (x) = W (x).

The following theorem uses the notations and results obtained in this chapter to show the existence and uniqueness of a solution to the discrete Dirichlet problem for Poisson’s equation on Z ³ which tends to zero as the argument tends to infinity.

Such a function may then be used as an analogue to the Green function on Z ³ . Theorem 4. Let w(a) be a given function on Z ³ such that P

a∈Z

|w(a)| < ∞.

Then the equation ∆ ₃ u = −w has a unique solution u(a) such that u(a) → 0 as kak → ∞.

Proof. (5.3) defines a continuous function W (x). Let us define (5.8) u(a) = (2π) ⁻³

Z Z Z

x

,x

,x

∈[−π,π]

e ^ia·x W (x)

P (x) dx 1 dx 2 dx 3 .

By the right hand side of (5.5) this integral is absolutely convergent. Then the Riemann–Lebesgue lemma states that the Fourier coefficients tend to zero as

|a| → ∞. Further operating on (5.8) with ∆ 3 gives ∆ 3 u = −w, thus for w such that P

a∈Z

|w(a)| < ∞, u satisfies the posed equation and also tends to zero as kak → ∞.

Now suppose u 1 (a) is another solution which tends to zero as kak → ∞. Then

v = u − u ₁ is discrete harmonic, and v → 0 as kak → ∞. Let A be the least

upper bound of v. If A > 0, then v = A at some point, say p. But then since

v is discrete harmonic we must have v = A everywhere. This contradicts v → 0

as kak → ∞. Therefore A ≤ 0. Analogously if B is the greatest lower bound

of v, then B ≥ 0. Therefore we must have v = 0 everywhere. This implies the uniqueness of u. This completes the proof.

We shall now use Theorem 4 to derive a formula for a discrete analogue of the Green function for Z ³ . Define the function g on Z ³ such that

(∆ ₃ g)(a) = −1, a = (0, 0, 0);

(5.9)

(∆ ₃ g)(a) = 0, a 6= (0, 0, 0);

(5.10)

g(a) → 0, as kak → ∞.

(5.11)

Denote ∆ ₃ g = w. Then by (5.3) the transform of w is W = 1. Thus by (5.8) one gets

(5.12) g(a) = (2π) ⁻³

Z Z Z

x

,x

,x

∈[−π,π]

e ^ia·x

P (x) dx ₁ dx ₂ dx ₃ , and by Theorem 4, this the unique solution of (5.9), (5.10), (5.11).

For a grid, R ⊂ Z ³ , with a set of interior nodes that form a convex proper subset of Z ³ a similar function may be constructed by iteration. Let g ₁ = C > 0 at the origin and g ₁ = 0 otherwise. Let g ₂ = C > 0 at the origin and at other interior points let it be the arithmetic mean of g ₁ . Iteration gives a sequence g ₁ ≤ g ₂ ≤ g ₃ ≤ · · · ≤ C which converges to a function g satisfying

∆ ₃ g(a) = −1, a = (0, 0, 0);

∆ ₃ g(a) = 0, a 6= (0, 0, 0);

g(a) = 0, a ∈ ∂R.

As an example consider the grid

Ω = {x = (x ₁ , x ₂ , x ₃ ) ∈ Z ³ ; 0 ≤ x ₁ ≤ 4, 0 ≤ x ₂ , x ₃ ≤ 2}.

Thus the interior points are (1, 1, 1), (2, 1, 1) and (3, 1, 1). Let g(1, 1, 1) = C = 2, g(x) = 0, otherwise. Then the iteration above corresponds to the following algoritm: g ₁ (2, 1, 1) = C/6 = 1/3, g ₁ (3, 1, 1) = 0

for i = 2 to 8,

g _i (2, 1, 1) = (g _i−1 (3, 1, 1) + 2)/6;

g _i (3, 1, 1) = (g _i−1 (2, 1, 1))/6.

The algoritm produces the following sequence for the vector (g _i (2, 1, 1), g _i (3, 1, 1)):

(0.333, 0), (0.333, 0.0556), (0.3426, 0.0556), (0.3426, 0.0571), (0.3428, 0.0571), (0.3428, 0.0571), (0.3429, 0.0571), (0.3429, 0.0571). This corresponds to the so- lution which in this simple case can be obtained by solving a system of two equations for two variables, giving (g _i (2, 1, 1), g _i (3, 1, 1)) = ( ¹² ₃₅ , ₃₅ ² ).

An important observation is that the integrand in the right hand side of (5.8) has a singularity at the origin. But by (5.5), P (x) behaves like kxk ² as kxk → 0.

Now kxk ^−p is locally integrable for p < n, where n is the dimension. This is true since

Z

kxk≤1

kxk ^−p dx = Z 1

0

(A _n−1 r ⁿ⁻¹ )r ^−p dr

where A n−1 is the surface area of the unit sphere in R ⁿ . This means that the integrand of (5.8) would not be integrable for n = 2. But an explicit Green function for n = 2 is derived by other methods in the next chapter. Consider the discrete Poisson’s equation on a finite grid R ⊂ Z ³ . Then the function

(5.13) u(ξ) = X

y∈R

g(y − ξ)u(y), ξ ∈ R,

where g is given by the limit of the above iteration, is bounded on the boundary nodes as well as the interior nodes and satisfies Poisson’s equation for all interior nodes. Because this solution is bounded and the grid is finite there exists a unique solution, H, to the discrete Dirichlet problem for the same grid such that u − H is a solution to the discrete Poisson’s equation which vanishes on the boundary nodes.

6 A discrete calculus for Z[i]

This chapter shall briefly present the foundations of a discrete calculus for the Gaussian integers, Z[i], with the purpose of using the defined operators to con- struct a discrete analogue of the Green function. The Laplace operator shall coincide with the definition from Chapter 1 for the case n = 2. Furthermore the construction shall be motivated by the proof of a discrete analogue of the fun- damental theorem of calculus for Z[i]. The construction is formulated by Deeter

& Gray (1974) which in turn follows the outline and ideas first presented by Ferrand (1944). After this description a discrete analogue of the Green function shall be presented for Z ² according to McCrae & Whipple (1940). Finally it will be discussed how this Green function may be used to construct a Green function for Z[i] equipped with the described discrete calculus. For h > 0, Deeter & Gray (1974) define the discrete complex plane with net width h as follows

(6.1) Π _h = {z ∈ C; z = ah, a ∈ Z[i]}.

A net with width h and the net for n = 2 defined by Phillips & Wiener (1923) are isomorphic to Z ² and include a parameter which allows investigation of asymp- totic behavior of defined operators on the sets. An even h-net and an odd h-net are defined by

Π _E = {z ∈ C; z = ch, c = a + ib, a + b even}, (6.2)

Π _O = {z ∈ C; z = ch, c = a + ib, a + b odd}.

(6.3)

For z 0 ∈ Π h the even (odd) h-square associated with z 0 , denoted by Q(z 0 ), is the square with the vertices z ₀ , z ₀ + h(1 + i), z ₀ + 2hi, z ₀ + h(i − 1). The h-square associated with z ₀ , denoted by S(z ₀ ), is the square with the vertices z ₀ , z ₁ = z ₀ +h, z 2 = z 0 + h(1 + i), z 3 = z 0 + hi. Denote the values of a function f defined at these vertices by f _k , k = 0, 1, 2, 3. For a connected set R ⊂ C the discrete region associated with R is

(6.4) R _h = {z ₀ ∈ Π _h ; z is a vertex of an h-square lying in R}.

An interior net point of a discrete region is a point, z, such that z ∈ R h is an interior point of R. The interior of R _h , denoted by I _h , is the set of all interior points, and the even and odd interior are defined as follows

I _E = {z ∈ R _h ∩ Π _E ; z + h(−1 + i) ∈ R _h } (6.5)

I _O = {z ∈ R _h ∩ Π _O ; z + h(−1 + i) ∈ R _h } (6.6)

A net path from a to b (a, b ∈ Z[i]) is the set

Γ = {(z 0 , . . . , z m ); z 0 = a, z m = b, z ∈ Π h , |z j−1 − z j | = h √ 2}.

If in this set every z ∈ Π _E (Π _O ) the term even (odd) net path is used. The lower left corner of the square determined by two successive points z j−1 , z j is denoted z _j−1 ^∗ . A closed net path has the property a = b. The discrete boundary of R _h , denoted by ∂R _h , is the closed path or paths that lie on the boundary of R. If the boundary of R is a simple closed curve, i.e., a curve with no endpoints that does not intersect itself, then ∂R _h is called simple discrete. The extended even and odd discrete regions associated with a region are defined by

R b _E = [

z∈I

{ξ ∈ Π _E ; ξ ∈ Q(z)}, (6.7)

R b _O = [

z∈I

{ξ ∈ Π _O ; ξ ∈ Q(z)}.

(6.8)

Deeter & Gray (1974) define discrete analogues of the operators _∂x ^∂ , _∂y ^∂ , _∂z ^∂ =

1 2

 ∂

∂x − i _∂y ^∂ 

, _{∂ ¯} ^∂ _z = ¹ ₂ 

∂

∂x + i _∂y ^∂ 

for a function defined on S(z ₀ ) as follows:

(6.9) ∆f ₀

∆x = f ₂ − f ₀ h √

2 , ∆f ₀

∆y = f ₃ − f ₁ h √

2 ,

(6.10) ∆

∆z = 1

√ 1 + i

 ∆

∆x − ∆

∆y



, ∆

∆¯ z = 1

√ 1 − i

 ∆

∆x + ∆

∆y

 .

Recall that f _k = f (z _k ), k = 0, . . . , 4, where z _k are the vertices of S(z ₀ ). A function is discrete analytic on S(z 0 ) iff

(6.11) ∆f ₀

∆¯ z = 0, which by (6.9) implies

∆f ₀

∆z = f ₂ − f ₀

h(1 + i) = f ₃ − f ₁ h(1 − i) (6.12)

∆f ₀

∆¯ z = ∆f ₀

(6.13) ∆z

Further Deeter & Gray (1974) define the even (odd) line integral along the even (odd) path Γ by

(6.14)

Z

Γ

f ∆z =

m

X

j=1

f (z ^∗ _j−1,j )(z _j − z _j−1 ).

The following discrete analogue of the fundamental theorem of calculus motivates the defined operators.

Theorem 5. Let f be a discrete analytic function on a simple discrete region b R _h . Suppose a, b ∈ b R _E ( b R _O ). Let Γ be an even (odd) net path from a to b. Then (6.15)

Z

Γ

∆f

∆z ∆z = f (b) − f (a).

Proof. For a given z _k−1 ∈ Γ = {z ₀ , . . . , z _m } there are four possibilities for z _k−1,k ^∗ . If z _k = z _k−1 + h(1 + i) then z _k−1,k ^∗ = z _k−1 and z _k − z _k−1 = h(1 + i) which gives ^{∆f (z}

∗ k−1,k

)

∆z (z _k − z _k−1 ) = ^{f (z}

_h(1+i) ^{)−f (z}

⁾ h(1 + i) = f (z _k ) − f (z _k−1 ) by (4.9). If z _k = z _k−1 + h(−1 − i) then z _k−1,k ^∗ = z _k−1 + h(−1 − i) and z _k − z _k−1 = h(−1 − i), which gives ^{∆f (z}

∗ k−1,k

)

∆z (z _k − z _k−1 ) = ^{f (z}

_h(−1−i) ^{)−f (z}

⁾ h(−1 − i) = f (z _k ) − f (z _k−1 ) by (4.9). Similarly z _k = z _k−1 + h(1 − i) and z _k = z _k−1 + h(i − 1) both imply

∆f (z _k−1,k ^∗ )

∆z (z _k − z _k−1 ) = f (z _k ) − f (z _k−1 ) using (4.9). Hence R b

a

∆f

∆z ∆z = P m

j=1 (f (z _j ) − f (z _j−1 )) = f (z _m ) − f (z ₀ ). This completes the proof.

Let f be defined on S(z _i ), i = 0, 1, 2, 3. Then the discrete Laplace operator is given by

4 ∆ ² f ₀

∆z∆¯ z = 4 ∆ ² f ₀

∆¯ z∆z =: D h f 2

(6.16)

which implies

D _h f ₂ = ∆ ² f ₀

∆x ² + ∆ ² f ₀

∆y ² , (6.17)

where

∆ ²

∆z∆¯ z = ∆

∆z

 ∆

∆¯ z



, ∆ ²

∆¯ z∆z = ∆

∆¯ z

 ∆

∆z

 ,

∆ ²

∆x ² = ∆

∆x

 ∆

∆x



, ∆ ²

∆y ² = ∆

∆y

 ∆

∆y

 . Recall that z ₂ = z ₀ + h(1 + i). Hence

(6.18) D h f 0 = 1

2h ² [f (z 0 + h(1 + i)) + f (z 0 + h(i − 1)) + f (z 0 + h(−1 − i))

+ f (z ₀ + h(1 − i)) − 4f (z ₀ )].

A function is discrete harmonic on R h iff

(6.19) D _h f (z) = 0, z ∈ I _h .

6.1 A discrete Green function for Z[i]

In this section a discrete analogue of the Green function for Z ² shall first be presented. Then it will be modified to give a function that may serve as a discrete Green function for Z[i] equipped with the discrete calculus described earlier in this chapter. McCrae & Whipple (1940) consider a rectangular grid R ⊂ Z ² consisting of (m + 2) rows (numbered from zero to m + 1) of (n + 2) nodes and seek a function v ₀ on the grid with the following properties

(∆ 2 v 0 ) (x) = 0, x ∈ R r {(a, b)}, (6.20)

(∆ ₂ v ₀ ) (a, b) = 1, (6.21)

where

(6.22) v ₀ (0, x ₂ ) = v ₀ (x ₁ , 0) = v ₀ (n + 1, x ₂ ) = v ₀ (x ₁ , m + 1) = 0.

What follows is a short presentation of the results given by McCrae & Whipple (1940) that constitute a solution to this problem. The function u ₀ (x ₁ , x ₂ ) = Ae ^x

^α+ix

^β , is discrete harmonic on the grid if cos β + cosh α = 2. This is shown by direct calculation, since

∆ 2 u 0 = 1

4 Ae ^ix

^α e ^iα + e ^−iα
+ 1

4 Ae ^x

^β e ^β + e ^−β
= 1

2 (cos α + cosh β) . Therefore the functions defined by

u ₁ (x) =

n

X

j=1

C(j) sin x ₂ jπ

n + 1 sinh x ₂ β _j sinh[(m + 1 − a)β _j ], x ₁ ≤ a, (6.23)

u ₂ (x) =

n

X

j=1

C(j) sin x ₂ jπ

n + 1 sinh x ₂ β _j sinh[(m + 1 − x ₁ )β _j ], x ₁ ≥ a, (6.24)

where C(j) ∈ R, cos _n+1 ^jπ + cosh β _j = 2, will satisfy the condition (6.20) when x ₁ = a, x ₂ 6= b, and (6.21) when x ₁ = b, x ₂ = a, if the constants C(j) are chosen such that

1 4

n

X

j=1

C(j) sin x ₂ jπ

n + 1 sinh x ₂ β _j sinh[(m + 1)β _j ] =

( 0, x ₂ 6= b

1

2 (n + 1), x ₂ = b.

Now the relation

n

X

j=1

sin x 2 jπ

n + 1 sin bjπ n + 1 = 1

2

n

X

j=1



cos (b − x 2 )jπ

n + 1 − cos (b + x 2 )jπ n + 1



=

( 0, x 2 6= b,

1

2 (n + 1), x 2 = a,

gives

(6.25) C(j) =

8

n+1 sin _n+1 ^bjπ sinh β _j sinh(m + 1)β _j . Inserting (6.25) into (6.23), (6.24) and letting n → ∞ gives

u 1 (x) = 8 π

Z π 0

sin bλ sin x 2 λ sinh[(m + 1 − a)µ]

sinh β _j sinh(m + 1)β _j dλ, (6.26)

u ₂ (x) = 8 π

Z π 0

sin bλ sin x ₂ λ sinh[(m + 1 − x ₁ )µ]

sinh β _j sinh(m + 1)β _j dλ, (6.27)

cos λ + cosh µ = 2.

(6.28)

This is shown by considering the function f = sin(c ₁ x) sin(c ₂ x), x ∈ [0, π], c ₁ , c ₂ ∈ Z. Let {I _j }, be an interval partition of I. For any  > 0, continuity of f implies that the oscillation of f in each interval, I _j , is less than , if the partition is fine enough. Then define the functions h ₁ , h ₂ as follows: for each interval let h ₁ = max f on the interior of the interval and h ₁ = f on the boundary of the interval, and let h ₂ = min f on the interior of the interval and h ₂ = f on the boundary of the interval. Recalling the measure of an interval, I ∈ R, (for integration in the Riemann sense), m(I) = [a ₁ , a ₂ ], a ₁ = min _x∈I x, a ₂ = max _x∈I x, this means R

I (h ₁ − h ₂ )dx < m(I), since R

I h ₂ dx ≤ R

I f dx ≤ R

I h ₁ dx. Hence if ξ _j ∈ I _j is an arbitrary point then

X

j

f (ξ j )m(I j ) → Z

I

f dx,

as the fineness of the partition tends to zero, i.e., the interval lengths decrease to zero. Now defining λ j = _n+1 ^jπ gives λ ∈ I = [0, π] for m ∈ N. Dividing I into n intervals of equal size, I _j , j = 1, . . . , n, where only the boundaries intersect, gives the measures, m(I _j ) = _n+1 ^π . Thus

1 π

π n + 1

X

j

sin x 1 λ j sin aλ j = m(I k ) X

j

f (λ j )→ R

I f (λ)dλ = 1 π

Z π 0

sin bλ sin x ₂ λdλ, as the interval lengths decrease to zero.

Now letting b → ∞, x ₂ − b = s < ∞ in (6.26), (6.27) gives u ₁ (x ₁ , s) = 4

π Z π

0

cos sλ sinh x 1 µ sinh(m + 1 − a)µ sinh µs sinh(m + 1)µ dλ, (6.29)

u ₂ (x ₁ , s) = 4 π

Z π 0

cos sλ sinh x ₁ µ sinh(m + 1 − x ₁ )µ sinh µs sinh(m + 1)µ dλ.

(6.30)

The above relations hold, since 8 π

Z π 0

cos((s + b)λ) sinh[(m + 1 − a)µ]

sinh β _j sinh(m + 1)β _j dλ =

 1

(b + s) 8

π sin((s + b)λ) sinh[(m + 1 − a)µ]

sinh β j sinh(m + 1)β j

 π 0

→ 0, when b → ∞.

And similarly for u 2 . Letting m → ∞ in (6.29), (6.30) gives u ₁ (x ₁ , s) = 4

π Z π

0

cos sλ sinh(x ₁ µ)e ^−aµ sinh µ dλ, (6.31)

u ₂ (x ₁ , s) = 4 π

Z π 0

cos sλ sinh(x ₁ µ)e ^−x

^µ sinh µ dλ.

(6.32)

This is seen by rewriting the trigonometric expressions on exponential form.

sinh[µ(m + 1 − a)]

sinh µ(m + 1) = e ^µ(m+1−a) − e ^{−µ(m+1−a)} e ^µ(m+1) − e ^−µ(m+1) =

e ^−µa − e −2µ(m+1)a+µa

1 − e ^−2µ(m+1) → e ^−µa , as m → ∞.

Consider now

u ₁ (a, b) − u ₁ (x) = 2 π

Z π 0

sinh(x ₁ µ)e ^−aµ − cos(sλ) sinh(aµ)e ^−x

^µ

sinh µ dλ, x ₁ ≥ a.

Letting a tend to +∞ while keeping x ₁ − a = r finite gives u 1 (a, b) − u 1 (x) → 2

π Z π

0

1 − cos(sλ)e ^−rµ

sinh µ dλ, r ≥ 0, and similarly for u ₂ ,

u ₂ (a, b) − u ₁ (x) → 2 π

Z π 0

1 − cos(sλ)e ^rµ

sinh µ dλ, r ≤ 0.

Now in (6.32), (6.31), u ₁ (x) = u ₂ (x), x ₁ = a. Define u = u ₁ , x ₁ ≤ a, u = u ₂ , x 1 ≥ a. Then

(6.33) u(a, b) − u(x) → 2 π

Z π 0

1 − cos(sλ)e ^−|r|µ

sinh µ dλ = : ν(r, s),

where cos λ + cosh µ = 2, is a bounded function on the grid. Several properties of ν(r, s) should be noted. In the derivations the rows and columns of the infinite grid are equivalent and a consequence of this is the symmetry of ν(r, s), i.e.,

(6.34) ν(r, s) = ν(s, r).

A rigorous proof of this fact is given by McCrae & Whipple (1940).

Furthermore

∆ ₂ ν(r, s) = 2 π

Z π 0

4 − (cos(s + 1)λ + cos(s − 1)λ) e ^−|r|µ

sinh µ dλ

+ 2 π

Z π 0

cos(sλ) e ^−|r+1|µ + e ^−|r−1|µ

sinh µ dλ − 4ν(r, s).

If r 6= 0, the relations

cos(s + 1)λ + cos(s − 1)λ = 2 cos(sλ) cos λ, e ^−|r+1|µ + e ^−|r−1|µ = 2e ^−|r|µ cosh µ = 2e ^−|r|µ (2 − cos λ)

imply that ∆ ₂ ν(r, s) = 0. In view of the symmetry we obtain ∆ ₂ ν(r, s) = 0 also if s 6= 0. Thus ν is harmonic at every point except the origin.

At the origin

∆ ₂ ν(0, 0) = 2 π

Z π 0

(1 − cos λ) + (1 − cos(−λ)) + 2(1 − e ^−µ )

sinh µ dλ − 4ν(0, 0)

= 8 π

Z π 0

1 − cos λ sinh µ dλ, using the symmetry (6.34). To evaluate the last integral, we introduce a variable ψ defined by

1 − cos λ = cosh µ − 1 = 2 cos ψ, so that

sin λ = 2 p

2 cos ψ sin ψ

2 and sinh µ = 2 p

2 cos ψ cos ψ 2 , whence

1 − cos λ

sinh µ dλ = −dψ.

When λ goes from 0 to π, ψ goes from π/2 to 0. Thus using the fact that ν(0, 0) = 0 together with the symmetry of ν(r, s) one gets

∆ ₂ ν(0, 0) = 4ν(1, 0) = 8 π

Z π 0

1 − cos λ

sinh µ dλ = − 8 π

Z 0 π/2

dψ = 4.

Finally the integral in the expression for ν(r, s) converges for all values of r, s ∈ Z.

Thus ¹ ₄ ν(r, s) is a discrete analogue of the Green function for Z ² and given a finite grid, Ω ⊂ Z ² , and a function f ∈ ` ^∞ defined on the interior points, the function

u(ξ) = X

z∈Ω

ν(z − ξ)

4 f (z), ξ ∈ Ω

is bounded and satisfies ∆ ₂ u(ξ) = f (ξ), ξ ∈ Ω ^◦ . Addition of an appropriate

bounded solution to the Dirichlet problem for the same grid yields a solution to

the discrete Poisson equation for Ω ⊂ Z ² and also for the grid Z ² . Now if the

aim is to find a function on Z[i] equipped with the discrete calculus described

by Deeter & Gray (1974) one may proceed as follows. Let z = ¹ _h (x 1 + ix 2 ), ξ = ¹ _h (u + iw), z, ξ ∈ Π _h , x ₁ − u = r, x ₂ − w = s, and consider the function

(6.35) G(z, ξ) = 2

π Z π

0

1 − cos ^s+r ₂ ^λ _h
e ⁻

sinh µ dλ.

Direct calculation gives

D _h G(z, ξ) = 4G(z, ξ) − G(z, ξ){2 cos λ + 2 cosh µ} = 0, z 6= ξ,

D _h G(ξ, ξ) = −2 2 π

Z π 0

(1 − cos λ) + (1 − e ^−µ )

sinh µ dλ = −4v(1, 0) = 4, z = ξ.

By (6.16)

∆ ² G(z, ξ − h(1 + i))

∆ξ∆ ¯ ξ = 1

4 D _h G(z, ξ) =

( 0, z 6= ξ, 1, z = ξ.

Thus given a finite grid, Ω ⊂ Z[i], and a function f ∈ ` ^∞ defined on the interior nodes, the function

u(ξ) = X

z∈I

G(z, ξ − h(1 + i))f (z), ξ ∈ Ω ⊂ Z[i],

belongs to ` ^∞ and satisfies ^∆ _{∆ξ∆ ¯}

^u(ξ) _ξ = f (ξ), ξ ∈ I _h . Addition of an appropriate bounded solution to the Dirichlet problem for the same grid yields a solution to the discrete Poisson equation for Z[i]. This solution also works for the grid Z[i].

Alternatively if one desires to use D _h directly the function u(ξ) = X

z∈I

1

4 G(z, ξ)f (z), ξ ∈ Ω ⊂ Z[i], solves the discrete Poisson equation.

7 A more general Laplace operator and an ana- logue to the method of Perron

This chapter shall generalize the theory of discrete harmonic functions on grids.

This shall be done by introducing a more general Laplace operator with the aid of weight functions. The definitions and results of this chapter are all presented thoroughly by Kiselman (2005). Let X be an arbitrary set. A complex-valued function g is called a structural function on X if it is defined on X × X and if the set {y ∈ X : g(x, y) 6= 0} is finite for all x ∈ X. A weight function is a structural function λ such that λ : X ² → R, λ ≥ 0 and P

y λ(x, y) > 0 for every x ∈ X. A weight function λ is normalized if P

y∈X λ(x, y) = 1 for all x ∈ X.

Kiselman (2005) defines the Laplacian of a function f at a point x as follows

∆f (x) = X

y∈X

λ(x, y)(f (y) − f (x)), x ∈ X.

(7.1)

If λ is normalized the Laplacian may be written

∆f (x) = X

y∈X

(λ(x, y) − δ(x, y))f (y), x ∈ X.

(7.2)

This definition of the Laplacian implies that (I + ∆) is increasing, thus f ≤ g implies (I + ∆)f ≤ (I + ∆)g. A function f : X → R is harmonic at x (in X) if

∆f (x) = 0 (for all x ∈ X respectively). The function f is called subharmonic at x (in X) if ∆f (x) ≥ 0 (for all x ∈ X respectively).

Given a weight function λ on a set X the boundary of X, denoted ∂X, is defined as {x ∈ X : λ(x, y) = 0 for all y 6= x, y ∈ X}. The interior of X is the complement of the boundary, X ^◦ = X r ∂X. Given a point a ∈ X a point y ∈ X such that λ(a, y) > 0 is called a neighbor of x. We define N ⁰ (a) = {a} and then inductively

N ^k+1 (a) = {y : λ(x, y) > 0 for some x ∈ N ^k (a)}, k ∈ N.

The union of all the N ^k (a), denoted C(a), is called the λ-component of a. X is boundary connected if C(a) intersects ∂X for all a ∈ X ^◦ . X is connected if C(a) = X for all a ∈ X ^◦ .

Proposition 1. If X is finite and boundary connected, then sup _X u = sup _∂X u for all subharmonic functions u on X.

Proof. Let u(a) = sup _X u = A, a ∈ X. Then u must take the value A at all points in N ¹ (a). Continuing this argument u must take the value A at all points in C(a). But by hypothesis C(a) intersects the boundary. This completes the proof.

Now the definitions of this chapter may be used to study the following formu- lation of the Dirichlet problem for Poisson’s equation

(7.3) ∆u(x) = f in X ^◦ , u = g on ∂X,

where f is a given function on X ^◦ such that f ≥ 0 and g is a given function on

∂X. The following theorem uses a method that is analogous to the method of Perron to deal with the case of the Dirichlet problem (7.4) for a finite set.

Theorem 6. Let X be a finite set and λ a weight function on X. Assume that X is boundary connected and assume that there exists a subsolution to the Dirichlet problem (7.4), i.e., that there exists a function w such that ∆w ≥ f in X ^◦ and w ≤ g on ∂X. Further assume that f ≥ 0. Then (7.4) has a unique solution.

Proof. First it will be shown that under the hypothesis of the theorem the unique solution is given as the supremum of all subsolutions. Define the function u as

(7.4) u(x) = sup

w

(w(x); ∆w ≥ f in X ^◦ , w ≤ g on ∂X).