MULTIDIMENSIONAL HYBRID BOUNDARY VALUE PROBLEM

The boundary value problems, considered in the paper, is a generalization of [24] in which the authors presented two-dimensional hybrids with mixed boundary value problems. Here we take a real Euclidean space R of dimension n on finite regions F ⊂ R that are polyhedral domains. The aim of this paper is to seek solutions of the Helmholtz equation with mixed boundary condition by analogy to two-dimensional cases. The solutions are presented as expansions into a series of special functions that satisfy required conditions at the (n− 1)-dimensional boundaries of F . The recent discovery of special functions [5, 10, 11, 16, 19, 23] makes realization of this idea easy and straightforward in any dimension. The new functions, called ’multidimensional hybrids’, satisfy the Dirichlet boundary condition on some parts of the boundary F and Neumann on the remaining ones. The methods used in the paper are the standard methods of separation of variables for differential equations (see for example [15, 18]) and the branching rule method for orbits of reflection groups (see for example [19, 24, 28]). The boundary value conditions play an important role in mathematics and physics. They are used, for example, in the theory of elasticity, electrostatics and fluid mechanics [4, 9, 26]. In §2, we present the well known Helmholtz equation and three types of boundary conditions. In § 3, we recall some facts about finite reflection groups. The next Section is devoted to special functions, projection matrices and branching rules. In § 5 we present 3D cases in details, namely B3, C3, C2×A1, G2×A1, A1× A1×A1. In the Appendix we list tables containing the values of functions on the boundaries of fundamental region.


Introduction
The boundary value problems, considered in the paper, is a generalization of [24] in which the authors presented two-dimensional hybrids with mixed boundary value problems.Here we take a real Euclidean space R n of dimension n on finite regions F ⊂ R n that are polyhedral domains.The aim of this paper is to seek solutions of the Helmholtz equation with mixed boundary condition by analogy to two-dimensional cases.The solutions are presented as expansions into a series of special functions that satisfy required conditions at the (n − 1)-dimensional boundaries of F .The recent discovery of special functions [5,10,11,16,19,23] makes realization of this idea easy and straightforward in any dimension.The new functions, called 'multidimensional hybrids', satisfy the Dirichlet boundary condition on some parts of the boundary F and Neumann on the remaining ones.The methods used in the paper are the standard methods of separation of variables for differential equations (see for example [15,18]) and the branching rule method for orbits of reflection groups (see for example [19,24,28]).The boundary value conditions play an important role in mathematics and physics.They are used, for example, in the theory of elasticity, electrostatics and fluid mechanics [4,9,26].
In §2, we present the well known Helmholtz equation and three types of boundary conditions.In § 3, we recall some facts about finite reflection groups.The next Section is devoted to special functions, projection matrices and branching rules.In § 5 we present 3D cases in details, namely B 3 , C 3 , C 2 ×A 1 , G 2 ×A 1 , A 1 × A 1 ×A 1 .In the Appendix we list tables containing the values of functions on the boundaries of fundamental region.

Helmholtz equation and boundary conditions
In this paper we consider the partial differential equation called the homogeneous Helmholtz equa-tion [15,18,25, and references therein]: where w-positive real constant, x = (y 1 , . . ., y n ) is given in Cartesian coordinates and ∆ = Using a standard method of separation of variables for (1) (see for example [15]) and searching for the solutions in the form Ψ(x) = X 1 (y 1 ) • • • X n (y n ), we have the following differential equation By introducing −k 2  1 , . . ., −k 2 n so-called separation constants, we get the solution of (2) in the form X 1  1 (y 1 ) = cos(k 1 y 1 ), . . .

Three
where ∂F = ∂F 0 ∪∂F 1 and f 0 , f 1 are given functions, defined on the appropriate boundary.

Finite reflection groups
Our method is general and can be presented for any crystallographic finite reflection groups G of any rank and any dimension which are associated with simple and semisimple Lie algebras/groups [1,7,10,27].
There is a complete classification of finite reflection groups given by Dynkin diagrams [2,3,10].These graphs provide the relative angles and relative length of the vectors of a set of simple roots of the root systems.There are two kinds of root systems according to the number of roots with different lengths: systems with one root length, and systems with two root lengths.A reflection r in a hyperplane orthogonal to the long/short root and passing through the origin of R n be denoted by r l /r s respectively.Working with finite reflection groups, it is convenient to use four bases in R n , namely natural e-, the simple root α-, co-root α-and weight ω-bases [2,7,10].The co-root basis α is defined by the formula αi = 2α i α i |α i .
The ω-basis is dual to simple root basis.The relationship between considered bases is standard for group theory and is expressed by There are two types of fundamental region either simplex for simple Lie group G or prism for semisimple one.The simplex with n + 1 vertices has the following coordinates where q i , i = 1, . . ., n, called co-marks, can be found in [6,10] for any simple Lie group G of any rank and any dimension.The fundamental region for prisms can be given in the following sense.
Let ∂F i be contained in the hyperplane generated by a set of orthogonal reflections r 0 , r 1 , . . ., r i−1 , r i+1 , . . ., r n , i = {0, . . ., n}, where r 0 is an affine reflection (it corresponds to long reflection).If r i corresponds to the reflection orthogonal to the short/long root then we denote a part of the boundary by ∂F s or ∂F l respectively.In other words we can say that the boundary ∂F of the fundamental region F will be denoted by ∂F l /∂F s if its normal vector is perpendicular to the long/short root α respectively.

Special functions as a solution of Helmholtz equation
There are four kinds of special functions of interest to us whose orthogonality on lattice fragment F is known for any simple Lie group [5,6,10,16,17,19,20,23, and references therein].The general formula for special functions (called orbit functions) [10,11] corresponding to the finite reflection group G is given by w∈G σ(w)e 2πi wλ|x , λ ∈ P + , x ∈ F (4) where the summation extends over the whole group G, P + denotes the set of dominant weights [10] and σ(w) = ±1 depends on the type of the orbit function.The homomorphism σ : G → {±1} is a product of σ(r l ), σ(r s ) ∈ {±1}.There are four types of maps σ [16,17]: All four families of functions defined above are formed as finite sums of exponential terms.The first two families, namely C-and S-functions are generalized cosine and sine functions.They are symmetric and skew-symmetric with respect to the finite reflection group [6,10,16,[19][20][21]23].The other two, S s -and S l -functions [11,12,16,17,23] have analogous properties as C-and S-functions.The main difference between them is their behaviour at the boundary of their domain of orthogonality in R n .Every finite group G generated by reflections can be reduced to a subgroup A 1 × • • • × A 1 using a branching rule method described in [13,14,19,22,24,28].This method allows us to do the separation of variables for special functions (5) corresponding to group G.As a result, we have all the functions written as a product of sine and cosine functions.
Remark 1.All four families of functions (5) presented above are solutions of the Helmholtz equation (1) where w 2 = 4π 2 λ|λ with one of the three types of boundary conditions described in § 2.
Projection matrix reduces any n-dimensional group G to a subgroup A 1 × . . .× A 1 [13,19].The branching rule allows one to divide any orbit of group G into a union of orbits of group A 1 .As an example see 3D cases described in § 5.

Remark 2. The union of orbits which we get after reduction determine our choice of separating constants used in solution of Helmholtz equation (1).
The behaviour of the functions C, S, S s and S l on the boundary ∂F can be summarize in the Tab. 1.

D N
Behaviour of the functions C, S, S s and S l on the boundary ∂F for any finite refleciton group G where * denotes any function non-equivalent to 0.
For any group G considered in the paper Cfunctions fulfil the Dirichlet condition with value nonequivalent to 0 and the Neumann condition with 0 value on the whole boundary.The S-functions behave inversely.The S s -functions fulfil the Dirichlet condition with a value non-equivalent to 0 on the part of boundary denoted by ∂F l and the Neumann condition with a value non-equivalent to 0 on the part of the boundary denoted by ∂F s .The S l functions behave inversely.In the case of C-functions we talk about Dirichlet boundary condition and S-functions -Neumann boundary condition.For S s -and S l -functions we talk about mixed boundary condition.In the next section we present 3D cases in details.

3D finite reflection groups
The 3 dimensional groups which we considered here are [2,7,8,10].We use the following notation for coordinates: where indexes e, ω, and α denote natural-, ω-, and αbasis, respectively.The action of the Laplace operator ∇ on the functions given in different bases can be found in [10].In the next subsections we describe each case in details.
For each case we present functions which are the solutions of Helmholtz equation (1).We give the exact forms of the projection matrices and branching rules which allow us to choose the separation constants used in (3).All functions described below fulfil one of the three types of boundary conditions described in §,2.

B 3 and C 3 groups
The α-basis vectors in Cartesian coordinates are As one can easily notice the short root for B 3 is α 3 and for C 3 are α 1 , α 2 .The fundamental regions F for B 3 and C 3 groups, written in ω-basis, have the vertices:  The reduction of B 3 and C 3 to a subgroup A 1 × A 1 × A 1 is given by the projection matrices Then the branching rule is the following: O(a, b, c) O(a, b, c)

−−→ O(a+b+c)O(b+c)O(c) ∪ O(b+c)O(a+b+c)O(c) ∪ O(a+b+c)O(c)O(b+c) ∪ O(b+c)O(c)O(a+b+c) ∪ O(c)O(a+b+c)O(b+c) ∪ O(c)O(b+c)O(a+b+c).
According to Remarks 1 and 2 the separation constants for B 3 and C 3 group we can choose as where where The explicit forms of orbit functions for B 3 and C 3 group have form + S c (x 1 )S a+b+c (x 2 )S b+c (x 3 ) + S b+c (x 1 )S c (x 2 )S a+b+c (x 3 ) + S c (x 1 )S b+c (x 2 )S a+b+c (x 3 ).
The functions on the right side of the above equations are special functions corresponding to group A 1 where The functions C µ (x i ) and S µ (x i ) are the solutions of Helmholtz equation (1) in the form (3) in 1D case.
For group B 3 the functions C-and S l -are real valued and S-and S s are purely imaginary.In the case of C 3 , the functions C-and S s -are real valued and S-and S l are purely imaginary.
The normal vectors shown on Fig. 2 are

C
The α-basis vectors in Cartesian coordinates have the form The vertices of the fundamental regions F for C 2 × A 1 , G 2 ×A 1 groups, shown in Fig. 3, written in ω-basis are The groups C 2 × A 1 , G 2 × A 1 can be reduced to a subgroup A 1 × A 1 × A 1 using a branching rule method described in [19,24].
The projection matrices and the branching rules are
For group C 2 × A 1 the functions C-and S s -are real valued and S-and S l are purely imaginary.In the case of G 2 × A 1 , the functions C-and S l -are real valued and S-and S s are purely imaginary.The normal vectors shown in Fig. 4 are 2 , 0}, n 5 = {0, 0, 1}, n 5 = {0, 0, 1}.
In the case of C 2 × A 1 the group normal vector n 3 is perpendicular to the short simple root.The rest of them, namely n 1 , n 2 , n 4 , n 5 are perpendicular to the long simple roots.So the boundaries that correspond to normal vectors are ∂F s for n 3 and ∂F l for the others.The values of the functions on the boundaries are summarized in Appendix in Tab. 4. In the case of G 2 × A 1 , the normal vector n 2 corresponds to the short simple root so to the boundary ∂F s and the rest of normal vectors to the long simple roots i.e. to the boundaries ∂F l .The values of the functions on the boundaries are given in Appendix in Tab. 5.

A
Although the root system of A 1 × A 1 × A 1 does not have two different lengths of roots, it is still an interesting case for us.The α-basis vectors in Cartesian coordinates have the form According to ( 4) and ( 5) there are two families of special functions C and S. By the analogy to homomorphism ( 5) we can define new families of functions.
where CCC, SSS correspond to C and S-functions, respectively and the rest of them to S l -and S s -functions.
All families of functions defined on the fundamental region fulfill mixed boundary condition (see Tab. 6).The projection matrix is the identity matrix and then the choice of separation constants is trivial:  O(a, b, c) where C µ (x i ), S µ (x i ) for i = 1, 2, 3 are the as in the previous cases.The first four families of functions are real valued and the rest of them are pure imaginary.Normal vectors shown on Fig. 5 are The values of the functions on the boundaries are shown in Appendix in Tab. 6.

Appendix
In Tables 2-6 we collect the values of special functions on the boundaries of the fundamental region F for each of 3D finite reflection groups presented in the paper.

no. /
Multidimensional Hybrid Boundary Value Problem Table 2.The values of C-, S-, S l -and S s -functions on the boundaries of fundamental region F of B3.The separation constants ki, i = 1, 2, 3 are given by ( 6) in § 5.1.
types of boundary conditions.D: A Dirichlet boundary condition defines the value of the function itself Ψ(x) = f (x), for x ∈ ∂F, where f (x) is a given function defined on the boundary.N: A Neumann boundary condition defines the value of the normal derivative of the function ∂Ψ ∂n (x) = f (x), for x ∈ ∂F, where n denotes normal vector to the boundary ∂F .M: A mixed boundary condition defines the value of the function itself on one part of the boundary and the value of the normal derivative of the function on the other part of the boundary and are shown in Fig.1.

Figure 1 .
Figure 1.The fundamental region F for B3 and C3 group.
) According to the branching rule x)Sc(z)+Sc(x)S