About the Dirichlet boundary value problem using Clifford Algebras

Abstract This paper reviews and summarizes the relevant literature on Dirichlet problems for monogenic functions on classic Clifford Algebras and the Clifford algebras depending on parameters on R3. Furthermore, our aim is to explore the properties when extending the problem to Rn and, illustrating it using the concept of fibres. To do so, we explore ways in which the Dirichlet problem can be written in matrix form, using the elements of a Clifford’s base. We introduce an algorithm for finding explicit expressions for monogenic functions for Dirichlet problems using matrices in R3. Finally, we illustrate how to solve an initial value problem related to a fibre.

1 Introduction 1.0.1 A higher-dimensional commutative structure and its disadvantages To define a product of vectors in R n+1 with n ≥ 2, we may consider ordinary polynomials in X 1 , ..., X n , where X j represents the x j -axis, j = 1, ..., n. The x 0axis represents the real numbers x 0 . Considering ordinary polynomials in X 1 , ..., X n does not imply distinguishing two terms, which are different only in the order of factors X j . By identifying squares X 2 j with real numbers −1, one obtains a finite-dimensional extension of R n+1 whose elements are finite linear combinations of the 2 n basis elements 1,X 1 , ..., X 1 X 2 , ..., X 1 X 2 , · · · X n . Since the order of the factors is not relevant, we then get an extension of R n+1 , where the product is commutative. A disadvantage of this approach is that the product of elements in R n+1 does not remain in R n+1 ; however, there is no alternative approach to avoid this issue. Moreover, in this extension the Laplace operator cannot be factorized in its two first order operators. If ∂ z is the differential operator 1 2 ∂ x + i∂ y and ∂ z , its conjugate, is defined by ∂ z = 1 2 ∂ x − i∂ y , then: Thus, the Laplace operator d 2 in the plane can be factorized by the two first order differential operators. Since such factorizations are useful for many applications, it will be important to generalize D from the Cauchy-Riemann operator by factorizing the Laplace operator in R n+1 . The construction above, which has a commutative product, does not have this property. First, observe that the rules X 2 j = −1 and X i X j = X j X i lead to a 2 ndimensional extension of R n+1 having the usual basis β n = {1, e 1 , ..., e n , e 1 e 2 , ..., e n−1,n , ..., e 1 e 2 · · · e n } with the identification X j = e j for j ∈ {1, 2, ..., n}. See [2,3].
The functions u = u(x) take values in the Algebra constructed above, that is, u(x) can be written as A u A (x)e A , where u A (x) are real-valued components of u(x) and e A ∈ β n . While holomorphic functions w = w(z) in the z-plane are defined by ∂ z w = 0, in higher dimensions it is necessary to define an analogous class of functions u = u(x), Du = 0, where D is the generalized Cauchy-Riemann operator defined by D = n i=0 e i ∂ j (2) where ∂ j represents the derivative with respect to x j . The conjugate Cauchy-Riemann operator D to D is given by Since e 2 i = −1 and e i e j = e j e i , we get where ∆ n+1 is the Laplace operator in R n+1 . Given that mixed derivatives appear in the latter formula, the Laplace operator cannot be factorized by the first order operators D and D. Moreover, since not all coefficients in (3) are real, the equation DDu = 0 is a coupled system for the real-valued components u A with solutions u = A u A e A of Du = 0. See [6,8].
Thus, we can now ask ourselves which structure is more useful for factorizing the Laplace operator. This is our motivation for the introduction of Clifford algebras

The usual Clifford algebra defined by equivalence classes
The starting point for the study of Clifford algebras are bilinear forms in linear spaces (see the basic monography [6] from F. Brackx, R. Delanghe and F. Sommen). However, the usual Clifford Algebras over R n+1 can also be constructed as equivalence classes of the n free independent variables X 1 , ..., X n , where one has to distinguish the two terms X µ 1 X µ 2 · · · X µ m with factors that are in different order. Equation (3) shows that it is possible to get a factorization of the Laplace operator provided e i e j + e j e i = 0 for each pair i j. Thus, a Clifford Algebra leading to the factorization ∆ n+1 = DD can be obtained if the formal structure is X 2 j + 1 and X i X j + X j X i , where i, j = 1, ..., n and i j.
We now are in a position to formally define Clifford Algebras. Consider n variables X 1 , ..., X n in a free algebra or free R-module, where the two products X µ 1 · · · X µ m are to be distinguished if the order of the factors is different (For instance, X 1 X 2 X 3 is said to be different from X 2 X 1 X 3 ).
Definition Two formal combinations, P and Q on R[X 1 , ..., X n ], are said to be equivalent if in their difference we can see one of the following terms: When two formal combinations, P and Q, are equivalent, we write P ∼ Q. Definition The Clifford Algebra A n is the set of all equivalence classes of formal combinations in R[X 1 , ..., X n ] with respect to the equivalence relation ∼.
Observe that in particular we have This means, in the language of equivalence classes, that This, however, is the same as

Remark 1
• Algebraic properties (such as the associativity of the algebraic operations and the distributive law) follow from the corresponding properties of polynomials. Note, however, that the multiplication is not commutative because X j X k = −X k X j for j k.
Example 1 In A 2 the structure relation is given by The basis is {e 0 = 1, e 1 , e 2 , e 12 }.
An element different from 0 is given by a = a 0 + a 1 e 1 + a 2 e 2 + a 12 e 12 .
Recently, Clifford analysis associated to new algebraic structures has attracted special attention. If one generalizes the structure used to define the Clifford Algebra, one obtains non-classical Clifford Algebra. Even though, classical Clifford Algebras have been sufficiently developed in the last decade, there is a need to study properties resulting from a more general structure, which allow for a better understanding of natural phenomena. [5,9,12,[18][19][20]22]

Clifford Algebras depending on parameters
Let p be any parameter running on certain set and, let α j (p) and γ ij (p) = γ ji be real-valued functions depending on that parameter p, where i, j = 1, ..., n and i j. Furthermore, k j ≥ 2 are natural numbers. For applications to partial differential equations in a domain Ω of R n+1 , one may assume that p is the variable x running in Ω.
Consider n free independent variables X 1 , X 2 , ..., X n over R n . Then, the vectors of R n+1 can be represented with a linear combination of the n free independent variables, where this identification preserves the linear structure. As before, one has to distinguish the two terms X µ 1 X µ 2 · · · X µ m with factors that are in different order. That way, one gets an infinite-dimensional extension of R n+1 in which, the product of vectors can be computed. In order to obtain only a finite-dimensional extension, we consider a equivalence class through the structures X k j j + α j (p) and X i X j + X j X i − 2γ ij (p), where i, j = 1, ..., n and i j.
Using this structure, each term of a formal linear combination of X 1 , ..., X n can be written as cX ν 1 1 · · · X ν n n , where c is a real constant and the exponents ν j are not greater than k j − 1, i.e., 0 ≤ ν j ≤ k j − 1. The Clifford algebra generated by the structure (5) will be denoted by A n (p | k j , α j (p), γ ij (p)) if n ≥ 2, and When the coefficients α j , γ ij do not depend on the parameter p, we write A n (k j , α j , γ ij ) and A 1 (k, α) resp. instead of (6). As usual, one denotes X j by e j , e 1 e 2 by e 12 and, so on. Then, for A n (p | k j , α j (p), γ ij (p)) has the basis e ν 1 1 e ν 2 2 · · · e ν n n , 0 ≤ ν j ≤ k j , j = 1, ..., n, and so we have, dim A n (p | k j , α j (p), γ ij (p)) = k 1 · · · k n , for n ≥ 2 and for n = 1 dim A 1 (p | k, α(p)) = k.
Finally, one important difference between classical Clifford Algebras and the Clifford-type Algebras is the second-order differential operator: See the papers [22,29,30] for more details.

Clifford valued function
Let Ω be an open and connected domain in R n+1 with points x = (x 0 , x 1 , ..., x n ). Furthermore, let u(x) be a function with values in A n (2, α j , γ ij ) defined on Ω.
Denoting the real-valued components of u(x) by u A (x), that is, Where A is a set of index combination. See [4,6,8,17,29].
Any Clifford-algebra-valued function u = A u A e A , which is monogenic with respect to the usual Cauchy-Riemann operator D and, the Clifford Algebra A n (2, α j , γ ij ) introduced above will lead to the homogeneous second-order differential equation: Remark 2 Hereinafter, we will use A n,2 =A n (2, α j , γ ij ) to refer to the case α j , γ ij constants and A * n,2 =A n (2, α j , γ ij ) to refer to no constant case.
2 Dirichlet Boundary value problem

Prescription of the values on the boundary
The imaginary part for a holomorphic function, v, with a bounded domain Ω in the complex plain is uniquely determined by its boundary values. Then, the Cauchy-Riemann system gives the real part, u, for the first order system, which in turn is completely integrable: Therefore, u is uniquely determined in simply connected domains up to a real constant. Thus, u is uniquely determined at any point on Ω or Ω, if applicable.
The system for u is completely integrable because u is a solution for the Laplace equation.
Akin to the complex case, a monogenic function, u, is continuously differentiable in an open connected set, Ω ⊂ R n+1 . The function's values belong to a Clifford Algebra A n , such that its 2 n real-valued components are related through the Cauchy-Riemann system in R n+1 , i.e. Du = 0, consisting of 2 n real first order partial differential equations in Ω. This condition again implies that it is not possible to arbitrarily and simultaneously prescribe boundary values for all the components. See [13,29,31].
Below, we will present a method to solve the Dirichlet boundary problem in a cylindrical domain in A 2,2 and A * 2,2 . Afterwards, we will present a way to extend the solution in A * n,2 using distinguished domains.

Cylindrical domain
Let Ω be a cylindrical domain whose closure Ω is given by where Ω 0 is a simply connected domain in the x 1 −, x 2 -plane and, ψ 1 and ψ 2 are continuously differentiable in Ω 0 . Then, the lower covering surface of Ω in the direction of the x 0 -axis is Theorem 1 The Dirichlet problem in Ω for a twice-continuous monogenic function with values in A 2,2 is uniquely solvable, where the Dirichlet problem can be described as: Find a monogenic function with respect to the Clifford Algebra A 2,2 , for which two components u 1 and u 2 are arbitrarily prescribed on the whole boundary ∂Ω, whereas u 12 has prescribed values on the lower base S 0 of the cylindrical domain Ω and u 0 is prescribed at only one point of Ω.
In [24,25,29], the authors solved a Dirichlet problem in a cylindrical domain for monogenic functions with values in A 2,2 .

Dirichlet boundary valued problem on
In this case, the differential second order equation is elliptic if γ 2 < αβ. The Cauchy-Riemann equation Du = 0 for a monogenic function u in A 2 (2, α, β, γ) is equivalent to the system for its four real-valued components u 0 , u 1 , u 2 and u 12 . Note that the parameters α, β and γ may depend on another variable, for example, x in R 3 .
Since the four components u 0 , u 1 , u 2 and u 12 of a monogenic function are solutions for the Laplace equation, u 1 and u 2 are uniquely determined by their boundary values for the entire boundary ∂Ω. It follows from equation (13) that knowing u 1 and u 2 , u 12 can be determined by integrating over the x 0 -direction, provided that the values for u 12 on S 0 are known (see the graphic illustration ??). Finally, one can use the remaining three equations (10)- (12) to calculate the component u 0 , splitting up the Cauchy-Riemann system in R 3 into two completely integrable first order systems for u 12 and u 0 , respectively. The component u 0 can be found from the system Since u 1 , u 2 and u 12 are solutions for the Laplace equation, the latter system for u 0 turns out to be completely integrable, that is ∂ k p j = ∂ j p k , k, j = 0, 1, 2.
Provided Ω is homotopically simply connected, u 0 is already uniquely determined by its value at one point P 0 of Ω. For more details on the solution, see [30]. Because α, β and γ may depend on the space-like component, another interesting case is when α, β and γ depend on x ∈ R 3 .

Cauchy-Riemann systems with non-constant coefficients
Now suppose that the parameters α, β and γ of the structured relations depend on the variable x. Then, one obtains the same system (10) -(13) for the components of a monogenic function, but now the coefficients depend on x. Then, the differential equations (9) for the components u 0 , u 1 , u 2 and u 12 are coupled, whereas they are uncoupled when the coefficients are constant. The main parts are always the same, namely ∂ 2 0 u j + α∂ 2 1 u j + β∂ 2 2 u j − 2γ∂ 1 ∂ 2 u j , j = 0, 1, 2, 12. In addition to the principal parts, the second order differential equations for u 12 , u 1 and u 2 have the following linear terms, respectively: A similar expression can be obtained for the differential equation of u 0 .
The differential equation for u 12 is always uncoupled because the additional linear terms contain only derivatives for u 12 . The equations for u 1 and u 2 are coupled only with one another, when the parameters α, β and γ do not depend on x 0 . The equations for u 1 and u 2 are completely uncoupled if α and γ depend only on x 1 and β depends only on x 2 .
A similar calculation shows that the system ∂ k p j = ∂ j p k , k, j = 0, 1, 2 is also compatible for the case of non-constant coefficients. If the differential equations for u 1 and u 2 are uncoupled, then theorem 1 is also true for non-constant coefficients. See [30].
The following step is to solve the Dirichlet boundary value problem in R n+1 . In order to generalize our method to a Ω ⊂ R n+1 , professor Tutschke suggested a generalization of the cylindrical domain, the distinguishing domain. The key idea is that a given domain Ω can be decomposed into fibres with small parts with certain properties.

Decomposition into µ-dimensional fibres
Consider the Euclidean space R n+1 with the coordinates x 0 , x 1 , ..., x n . Choose µ with 1 ≤ µ ≤ n + 1 and, choose µ indices k 1 , ..., k µ . Then, the intersections of a given (bounded) domain Ω with µ-dimensional planes of R n+1 is given by: Definition 2 We say that the µ-dimensional fibres in the k 1 , ..., k µ -direction of the given bounded domain Ω are the intersections of the Ω with µ-dimensional planes.
Definition 3 We say that Ω can be decomposed into µ-dimensional fibres in k 1 , ..., k µ -direction if there exists a (1 + n − µ)-dimensional part S k 1 ...k µ of ∂Ω having the following properties: • In each plane P , there exists at most one point of S k 1 ...k µ , which is a boundary point for the corresponding fibre.
• Ω is the product of S k 1 ...k µ with the closure of the fibres.
The subset S k 1 ...k µ is called the distinguishing part of the corresponding decomposition of Ω.
Example 2 Consider the cylindrical domain Ω ⊂ R 3 given in (??). Ω can be decomposed in the lower or upper covering surface, this is a distinguishing part for the decomposition in the x 0 -direction that represents the 1-dimensional fibre. See [26] for more details on fibres.
Remark 3 Using more general operators of Cauchy-Riemann type, the Dirichlet problem in the cylinder can also be solved, see [9,11].
In the following section, we are going to represent the Cauchy-Riemann system in matrix form in A * 2,2 , we will also discuss the matrix representation in A n and, show an example using a computer implementation for A 3 . See [7].
The Cauchy-Riemann operator is given by: Thus, the Cauchy-Riemann system is given by:

Matrix representation in A n
Using the matrix basis β n given in (14), we can get a matrix representation for the Dirac operator in A * n,2 where, E i,n are matrices R 2 n ×2 n and ∂ i represents the derivative with respect to Let Ω be an open and connected domain in R n+1 whose points will be denoted by x = (x 0 , x 1 , ..., x n ). Let, u(x) be a function with values in A n defined on Ω.
Denoting the real-valued components of u(x) by u A (x), that is, where A is a set of index combinations and E 0,n = I 2 n . See [7] for more details on the matrix construction for the operator D n We can represent the Cauchy-Riemann system as the product of two matrices:

Boundary value problem using fibres
Theorem 2 Consider a Clifford valued function u given by u( where e A ∈ {e 0 = 1, e 1 , e 2 , e 12 } such that u A ∈ AC 2 (Ω) Then, the Dirichlet problem in a Decomposition of Ω ⊂ R 3 in distinguish parts given by: • The lower covering surface, that is, a distinguishing part for the decomposition in the x 0 -direction representing the 1-dimensional fibre S 0 .
• S 01 , which is the distinguishing part in the x 0 , x 1 −directions, corresponding to the 2−dimensional fibre.
• S 012 , which is the distinguishing part x 0 , x 1 , x 2 − directions, corresponding to the 3−dimensional fibre.
for monogenic function u with values in R 3 as defined above, is uniquely solvable.
Proof: Consider the Cauchy-Riemann operator D 2 given by Then, the Cauchy-Riemann system D 2 u = 0 for an A 2 −valued function u = u 0 e 0 + u 1 e 1 + u 2 e 2 + u 12 e 12 is given by: By multiplying these matrices, we obtain four equations: First, two of these components have to be prescribed. Without loss of generality, suppose we prescribe arbitrarily u 1 and u 2 on the whole boundary ∂Ω, whereas u 12 has prescribed values on the lower covering surface S 0 and u 0 is prescribed at one point ofΩ. In order to solve the system (17) - (20) we consider the equations (17) - (19) as the following system for u 0 : where We will show that the right-hand sides of the system (21) are compatible. Consider the expression Using the Laplacian for u 1 and remembering that ∆u 1 = 0 then we can replace ∂ 2 0 u 1 + ∂ 2 1 u 1 by −∂ 2 2 u 1 in the previous expression. Doing this, we get: Following from equation (20), the latter expression is zero. Analogously, we use ∆u 2 = 0 to the expression ∂ 2 p 0 − ∂ 0 p 2 and use (20) again to conclude that ∂ 2 p 0 − ∂ 0 p 2 = 0. For ∂ 2 p 1 −∂ 1 p 2 , we have to use ∆u 12 = 0 to conclude that ∂ 2 p 1 −∂ 1 p 2 = 0. Thus, the system (21)  The Cauchy-Riemann system D 3 u = 0 for an A 3 −valued function u = u 0 e 0 + u 1 e 1 + u 2 e 2 + u 3 e 3 + u 12 e 12 + u 13 e 13 + u 23 e 23 + u 123 e 123 is given by: For the component u 0 , we have the system: where We will show that the right-hand sides of the system (22) are compatible. Consider the expression Using the Laplacian for u 13 , then we get: Finally, by rows 6 and 8 from the matrix for D 3 u = 0 we have that ∂ 3 P 1 − ∂ 1 P 3 = 0. Analogously, the remaining equations are equal to zero by using the Laplacian of certain components and certain rows of the matrix D 3 u = 0. We list them here: • ∂ 2 P 1 − ∂ 1 P 2 = 0 using the Laplacian for u 3 , and rows 4 and 8 from the matrix system.

Algorithm 3.1 (System's integrability)
Input: • The dimension n of the Clifford algebra A n . Output: • The 2 n−1 prescribed components and the 1−dimensional fibres for the system.
• The verification that there is a solution for the subsystem for the coupled components.
• The verification that there is a solution for the system D n u = 0.
Step 1: Construct the matrix representation for D n u = 0 Step 2: Identify the prescribed componentes.
Step 5: Verify the integrability of the subsystem for the (n − 1)-dimensional .
Step 6: Verify the integrability of the subsystem for u 0 .
Remark 5 With the last code we can solve any Dirichlet boundary value problem in R n with the proper fibre.
3.4 Explicit solution using the matrix representation, the A 2 case.
In this section we will show how we can use the matrix representation formulation to derive an explicit solution u for D 2 u = 0. After solving the Dirichlet boundary value problem, we get a remarkable result: an explicit solution for the Dirichlet boundary problem. For A 2 , we already know that we must prescribe 2-components.
Using the algorithm developed for this purpose (see appendix ??), the Cauchy Riemann system D 2 u = 0 is given by: Step 1: Compute the system ∂ x j u 0 = p j Step 2: Test u 1 , u 2 in the system Step 2: If ∂ x j u 0 = p 0 is false for j = 0, 1, 2, then this is the right equation in order to compute u 0 . Step 1: Compute the matrix D in terms of the matrices I, E 1,2 , E 2,2 , E 12,2 ∈ R 4×4 Step 2: Compute the matrix Du Step 3: Construct the compatibility equations from the system Du = 0.
Using the algorithm 3.3 (for more details see appendix) then the Cauchy Riemann system D 2 u = 0 is given by where dj, j = 0, 1, 2 represent the partial derivative in the respective direction. D 2 is the 4 × 4 matrix, given by: and u is given by: As we want to obtain solutions for the Cauchy-Riemann system, it is enough to consider u as a 4×1 matrix. Using the right compatibility condition we get the following explicit result: where a ∈ R.

Initial valued problem for monogenic initial data
We are interested in solving the initial value problem: in the space of monogenic functions satisfying the differential equations lu := Du − F(x, u) = 0, where F(x, u) is an anti-monogenic (Df = 0) function in Clifford Algebras depending on parameters and, the initial function ϕ is a generalized monogenic function satisfying a differential equation with a monogenic right-hand side.
The starting point is the classic Cauchy-Kovalevskaya problem for an evolution equation with a holomorphic right-hand side, where each initial value problem with holomorphic initial data is uniquely solvable. See [16,21,34].
The concept of associated spaces generalizes the connection between monogenic right-hand sides and monogenic initial data. If the initial data belong to an associated space of the right-hand side, then the initial value problem is uniquely solvable. Then, the desired solution of the initial value problem can be found as a fixed point (x * ) of a related operator U (x, t) = ϕ(x * ) + t 0 F(τ, x, u, ∂ j u)dτ. (26) In order to construct fixed points for this operator, we have to estimate the integrodifferential operator on the right-hand side of (26). Discussions on the estimate can be found in [21,23,27,33].
On the other hand, let u = u(x 0 , x 1 , ..., x n ) be the desired solution for a real-valued function with a completely integrable system in µ variables, 1 ≤ µ ≤ 1 + n: where ∂ k j means differentiation with respect to the variable x k j in an open and homotopically simply connected subset of the domain under consideration.
Using the concept of a cylindrical domain of order µ makes it possible to solve µdimensional systems (27) not only locally in µ-dimensional subsets; but also, globally in (1+n)-dimensional domains, in this case each fibre is defined by a uniquely determined point x * of S k 1 ...k µ . This point is the initial point of the curves γ of the particular fibre. Then, an explicit solution can be given by the line integral U (x) = ϕ(x * ) + γ µ j=1 p k j (ξ)dξ k j (28) where x * is a fixable chosen point and γ is any curve connecting x * and x (in the subset).
The integral does not depend on the special choice of γ because the system is completely integrable and the subset is homotopically simply connected by hypothesis. Finally, in each fibre the values of u are given by an integral (28) where γ is located in the corresponding fibre. More information related to the initial value problems can be found in [1,10,14,15,32].