## Abstract

In his deep and prolific investigations of heat diffusion, Lamé was led to the investigation of the eigenvalues and eigenfunctions of the Laplace operator in an equilateral triangle. In particular, he derived explicit results for the Dirichlet and Neumann cases using an ingenious change of variables. The relevant eigenfunctions are a complicated infinite series in terms of his variables.

Here we first show that boundary-value problems with simple boundary conditions, such as the Dirichlet and the Neumann problems, can be solved in an elementary manner. In particular, the unknown Neumann and Dirichlet boundary values can be expressed in terms of a Fourier series for the Dirichlet and the Neumann problems, respectively. Our analysis is based on the so-called global relation, which is an algebraic equation coupling the Dirichlet and the Neumann spectral values on the perimeter of the triangle.

As Lamé correctly pointed out, infinite series are inadequate for expressing the solution of more complicated problems such as mixed boundary-value problems. In this paper we show, further utilizing the global relation, that such problems can be solved in terms of *generalized Fourier integrals*.

## 1. Introduction

Solutions of *certain* linear elliptic boundary-value problems can be expanded in complete sets of eigenfunctions. Unfortunately, the actual form of these eigenfunctions is only known for simple geometries. In fact, only geometries that allow the separation of variables yield well-known expressions for the associated eigenfunctions. But what happens when the separation of variables does not apply? Is it possible to construct the spectral characteristics of a fundamental domain that does not fit any separable coordinate system? Some examples where this construction is possible are presented in the present work. The approach used here has its roots in the unified transform method for analysing both lineal and integrable nonlinear partial differential equations (PDEs) introduced in Fokas (1997).

A crucial role in this analysis is played by a certain equation coupling all boundary values, which was called the global relation in Fokas (1997). The concrete form of this equation for the equilateral triangle was given in the important work of Shanin (1997), where it was called a functional equation.

A general overview of the problems solved in this paper is presented in the sequel, where notations and some elementary formulae are included in order to facilitate understanding the new results. We study boundary-value problems for the Laplace, the Helmholtz and the modified Helmholtz equations in the interior of an equilateral triangle. These equations are three of the basic equations of classical mathematical physics. In particular, they arise as the reduction of several fundamental parabolic and hyperbolic linear equations. Furthermore, the specific boundary conditions discussed here cover most cases of physical significance.

### (a) Notation and useful identities

*z*will denote the usual complex variable and*α*will denote one of the complex roots of unity:(1.1)An overbar will denote complex conjugation, in particular will denote the*Schwarz conjugate*of the function*F*(*k*).The complex numbers(1.2)will denote the vertices of the equilateral triangle, and will denote the interior of the triangle. The length of each side is

*l*. The sides (*z*_{2},*z*_{1}), (*z*_{3},*z*_{2}), (*z*_{1},*z*_{3}) will be referred to as sides (1), (2) and (3), respectively.On each side we identify the positive direction as and the outward normal as , as in figure 1. The functions(1.3)will denote the function

*q*(*x*,*y*), as well as the derivative of*q*(*x*,*y*), along the outward normal respectively for side (*j*).*E*(*k*) and*e*(*k*) will denote the following exponential functions:(1.4a)(1.4b)Given that the numbers

*α*and satisfy the obvious relations(1.5)it is straightforward to obtain analogous relations for*E*(*k*) and*e*(*k*). For example, the last three equations in (1.5) imply(1.6)

### (b) Formulation of the problem

We will investigate the basic elliptic equations in the interior of the equilateral triangle *D*, namely, we will study the equation(1.7)where *q*(*x*,*y*) is a real valued function and *λ* is a real constant. The case of *λ*=0, *λ* being negative and *λ* being positive correspond to the Laplace, the Helmholtz and the modified Helmholtz equations, respectively. We will analyse the following problems.

The Dirichlet problem(1.8)

The oblique Robin problem(1.9)where

*β*and*γ*are real constants and sin*β*≠0. The sum of the first two terms of the left-hand side of this equation equals the derivative of*q*^{(j)}(*s*) in the direction making an angle*β*with the positive direction of the side (*j*). The Neumann and the Robin problems correspond to the following particular choices of*β*and*γ:*(1.10)The Poincaré type problem(1.11)where

*β*_{1}is a real constant so that sin*β*_{1}≠0.*β*_{2}and*β*_{3}satisfy sin*β*_{2}≠0 and sin*β*_{3}≠0 and are given in terms of*β*_{1}by the expressions(1.12a)and the real constants satisfy the relations(1.12b)and(1.12c)A particular case of such a Poincaré type problem, which is solved in detail, is the modified Helmholtz equation with Neumann values on sides (2) and (3) and with Robin values on side (1), where the constant*γ*is given by .

We assume that the functions *f*_{j} have sufficient smoothness and that they are compatible at the corners of the triangle. The case of boundary conditions which are discontinuous at the corners will be considered elsewhere.

### (c) The global relation

As mentioned earlier, the approach used here is based on the analysis of the global relation, which is the fundamental algebraic relation that couples the Dirichlet and the Neumann values around the perimeter of the triangle. This equation, first derived for the case of an equilateral triangle in Shanin (1997; see also Fokas 2001) is(1.13)where the exponential function *E*(*k*) is defined in equation (1.4*a*), and *Ψ*_{j} and *Φ*_{j} are the following transforms of the Neumann and Dirichlet boundary values(1.14a)(1.14b)for each *j*=1,2,3, and every complex *k*≠0.

From the general methodology introduced in Fokas (1997, 2001) implies that the global relation must be supplemented by its Schwarz conjugate, as well as by the four equations obtained from these two equations, by replacing *k* with *αk*, in equation (1.14*a*) and in equation (1.14*b*). We will refer to these six equations as the *basic algebraic relations*. In this paper we present two different techniques for solving these equations.

#### (i) Solutions via infinite series

For simple problems it is possible to compute the unknown boundary values by evaluating the basic algebraic relations at particular discrete values of *k*. This yields the unknown boundary values in terms of infinite series. The Dirichlet and the Neumann problems are examples of problems which can be solved using this technique.

We use the Dirichlet problem to illustrate this approach. In this case the functions *Φ*_{j} appearing in the right-hand side of the global relation (1.13) can be immediately computed in terms of the given boundary conditions *f*_{j}; thus, the global relation becomes a single equation for the three unknown functions . Multiplying this equation by *E*(i*αk*), and multiplying the Schwarz conjugate of the global relation by *E*(−i*αk*), we find the following two equations (where we have used the last two of the identities in equation (1.6)):(1.15)(1.16)In these equations *A*(*k*) and *B*(*k*) are known functions and .

For the general Dirichlet problem, we will supplement these two equations with the four equations obtained from these equations by replacing the *k* in equations (1.14*a*,*b*) with *αk* and , respectively. However, there exists a particular case for which it is sufficient to analyse only the above two equations. This is the *symmetric* Dirichlet problem, namely, the problem where the functions *f*_{j} are all the same, *f*_{j}=*f*,*j*=1,2,3. Then the Neumann values are also the same, and, hence, *Ψ*_{j}(*k*)=Ψ(*k*), *j*=1,2,3. Thus, equations (1.15) and (1.16) become two equations for the three unknown functions *Ψ*(*k*), and *Ψ*(α*k*). Hence, any two of them can be expressed in terms of the remaining one, for example, and *Ψ*(α*k*) can be expressed in terms of *Ψ*(*k*). In particular, by subtracting equations (1.15) and (1.16) we find(1.17)where *G*(*k*)=*A*(*k*)+*B*(*k*) is a known function. Equation (1.17) is a single equation for the two unknown functions and *Ψ*(*k*). However, by evaluating this equation at those values of *k* for which the coefficient of vanishes, that is, at *e*^{2}(*k*)=1, or *k*=*s*_{n},(1.18)it follows that *Ψ*(*s*_{n}) can be determined. Recalling the definition of *Ψ*(*k*) and evaluating equation (1.17) at *k*=*s*_{n}, we find(1.19)Thus, *q*_{N}(*s*) can be expressed as a Fourier series.

For the general Dirichlet problem of equation (1.8), the six basic algebraic relations couple the nine unknown functions . Thus, any six of them can be expressed in terms of the remaining three. In particular, it is shown in §3 that can be expressed in terms of by the equation(1.20)where *X*(*k*) is known. In spite of the fact that this equation is a single equation for four unknown functions, it yields all the three Neumann values , *j*=1,2,3. Indeed, by evaluating equation (1.20) at those values of *k* for which the coefficient of vanishes, at *e*^{6}(*k*)=1, or *k*=*k*_{m}, where(1.21)equation (1.20) yields(1.22)where *M*(*k*_{m}) is known. This equation, in contrast to equation (1.19), involves *three* unknown functions. However, equation (1.21) gives three times as many values for *m* as equation (1.18). By replacing *m* in equation (1.22) with 3*n*,3*n*−1 and 3*n*−2, respectively, and inverting the left-hand sides of the resulting equations, we find(1.23)Thus, by solving this system of three algebraic equations it follows that each one of the Neumann boundary values can be represented in terms of a Fourier series (see proposition 3.2).

The analysis of the oblique Robin problem (equation (1.9)) is similar. However, in general, the explicit values of *s*_{n} and of *k*_{m} cannot be found. The values *k*_{m} satisfy the transcendental equation(1.24)Thus, we now have instead of in equation (1.22), where *k*_{m} satisfies equation (1.24). In the particular case of the Neumann problem, *k*_{m} satisfies equation (1.21).

#### (ii) Solutions via generalized Fourier integrals

For more complicated problems, such as equation (1.11), the basic algebraic relations can be solved in terms of a generalized Fourier integral. This technique is generic in the sense that it can also be used for the solution of simple problems.

We use such a simple problem, namely the symmetric Dirichlet problem, to illustrate this approach. It is shown in §4 that the integral defining can be solved for *q*_{N}(*s*). For *λ*≥0, *q*_{N}(*s*) is given by(1.25)By replacing in this equation with the expression obtained by solving equation (1.17) for , it follows that *q*_{N}(*s*) involves a known integral, as well as an integral containing the unknown function *Ψ*(*k*). However, using the analytic properties of the integrant of the latter integral, it can be shown that this integral can be computed in terms of residues. Furthermore, these residues can be explicitly calculated in terms of the known function *G*(*k*). The situation for more complicated problems is similar: the unknown boundary values can be expressed in terms of known integrals, as well as integrals containing the three unknown functions . By exploiting the analytic properties of the integrants of the latter integrals, it can be shown that these integrals can be explicitly computed.

### (d) Integral representations for *q*(*x*,*y*)

When both the Dirichlet and the Neumann boundary values are known, the solution *q*(*x*,*y*) can be determined either using the classical integral representation in terms of Green's functions (Dassios & Kleinman 2000), or using the novel integral representations constructed in Fokas (2001) and Fokas & Zyskin (2002). For completeness, both representations are presented in §5.

### (e) Organization of the paper

In §2 we derive the global relation (1.13). In §3 we solve the symmetric Dirichlet problem (proposition 3.1), the general Dirichlet problem (proposition 3.2) and the general Neumann problem (proposition 3.3) and discuss the oblique Robin problem. In §4 we discuss the basic algebraic relations associated with the Poincaré boundary condition in equation (1.11) and derive the relations (1.12*b*) and (1.12*c*). In §5 we obtain an alternative representation for the symmetric Dirichlet problem and analyse the problem defined by equations (1.11) and (1.12*a*,*b*,*c*). A particular case of this problem, which is solved in detail in proposition 5.1, is a mixed boundary-value problem for the modified Helmholtz equation. In §6 we discuss the associated integral representations for *q*(*x*,*y*). Further discussion of these results is presented in §7.

## 2. The global relation

Writing the basic elliptic equation (1.7) with the complex variables we find(2.1)It is straightforward to verify that this equation can be rewritten in the form(2.2)where for the rest of this section (Fokas 2001). Suppose that equation (2.1) is valid in a simply connected bounded domain with a piecewise smooth boundary ∂*Ω*. Then, equation (2.2) and the complex form of Green's theorem imply(2.3)In the particular case where *Ω* is the triangular domain *D*, equation (2.3) becomes(2.4)where the function is given by the following line integral along the side (*j*) of the equilateral triangle(2.5)In what follows we will show that(2.6)where the functions *ρ*_{j}(*k*) are defined in terms of the functions *Φ*_{j}(*k*) and *Ψ*_{j}(*k*) by the equation(2.7)For this purpose we will use the following local parametrizations.

**Side 1.** On side (1) the variable *z* can be parametrized as(2.8a)Then *z*(−*l*/2)=*z*_{2} and *z*(*l*/2)=*z*_{1}. Since the normal and the tangential derivatives are parallel to the *x* and to the *y* axes, respectively, it follows that(2.8b)

**Side 2.** If *z* varies along side (2) and *ζ* varies along side (1), then . Thus,(2.9a)Note again that *z*(−*l*/2)=*z*_{3} and *z*(*l*/2)=*z*_{2}. The equation implies that(2.9b)

**Side 3.** In analogy with equations (2.9*a*) and (2.9*b*), if *z* varies along side (3) we find the equations(2.10a)and(2.10b)Finally, *z*(−*l*/2)=*z*_{1} and *z*(*l*/2)=*z*_{3}. Using equations (2.8*a*,*b*)–(2.10*a*,*b*) in the expressions in equation (2.5) we find equations (2.6) and (2.7).

## 3. The analysis of the global relation for simple boundary-value problems

### (a) The symmetric Dirichlet problem

We first give the details for the symmetric problem. In this case(3.1)where the function *Ψ*(*k*) is defined in terms of the unknown function *q*_{N}(*s*) by equation (1.14*a*) (without the superscript (*j*)), and the function *F*(*k*) is defined in terms of the given boundary condition *f*(*s*) by equation (1.14*b*), that is, by the equation(3.2)Using equations (3.1), the global relation (1.13) and its Schwarz conjugate yields equations (1.15) and (1.16), withHence, since *G*=*A*+*B*,(3.3)In summary, we have derived the following result:

*Let the real valued function q*(*x*,*y*) *satisfy equation* *(1.7)* *in the triangular domain D,* *with the Dirichlet conditions of equation* *(1.8),* *where*(3.4)*and the function f*(*s*) *is sufficiently smooth and satisfies the continuity condition f*(−*l*/2)=*f*(*l*/2). *Therefore*, *the Neumann boundary values are the same,* (*j*=1,2 *and* 3)*,* *and are given by the Fourier series*(3.5)*where s*_{n} *is defined by equation* *(1.18)* *and G*(*s*_{n}) *is given in terms of f*(*s*) *by equations* *(3.2)* *and* *(3.3)*.

### (b) The general Dirichlet problem

The global relation and its Schwarz conjugate yield equations (1.15) and (1.16), where the known functions *A*(*k*) and *B*(*k*) are now given by the equations(3.6)where(3.7)By replacing *k* in equations (1.15) and (1.16) with and then eliminating from the resulting two equations we find(3.8)Taking the Schwarz conjugate of this equation (or, equivalently, eliminating *Ψ*_{1}(*αk*) from the equations obtained from equations (1.15) and (1.16) by replacing *k* with *αk*) we find(3.9)Substituting *Ψ*_{2}(*αk*) from equation (3.8) and from equation (3.9) into equation (1.16), we find an equation involving *Ψ*_{3}(*αk*), and . Eliminating *Ψ*_{3}(*αk*) from this equation and from equation (1.15) we obtain equation (1.20) with *X*(*k*) given by the following equation:(3.10a)Letting *k*=*k*_{n}, we find(3.10b)Solving the algebraic equations (1.23) we find the following result.

*Let the real valued function q*(*x*,*y*) *satisfy equation* *(1.7)* *in the triangular domain D,* *with the boundary conditions of equation* *(1.8),* *where the given functions f*_{j}(*s*) *have sufficient smoothness and are continuous at the vertices*. *Then,* *the Neumann data* (*j*=1,2 *and* 3) *can be expressed in terms of the given Dirichlet data with the Fourier series*(3.11)*where k*_{m} *is defined by equation* *(1.21)**,*(3.12)*and*(3.13)*and X*(*k*_{m}) *is defined in terms of f*_{j}(*s*) *with equations* *(3.7)* *and* *(3.10a*,*b)*.

### (c) The oblique Robin problem

Suppose that *q*(*x*,*y*) satisfies the Poincaré boundary condition of equation (1.11), that is,(3.14)Substituting this expression into the definition of *ρ*_{j}(*k*), that is, into equation (2.7), we obtainIntegrating by parts we find the following expression for *ρ*_{j}(*k*):(3.15)where the function *H*_{j}(*k*) is defined by(3.16)the function *F*_{j}(*k*) is defined in terms of the given boundary conditions *f*_{j}(*s*) by the equation(3.17)the function *C*_{j}(*k*) involves the values of *q*(*x*,*y*) at the vertices,(3.18)and the function *Y*_{j}(*k*) involves the unknown Dirichlet boundary values,(3.19)In equations (3.15)–(3.19), *k* is complex and *k*≠0.

In the particular case of the oblique Robin problem (1.9), *β*_{j}=*β* and *γ*_{j}=*γ* (*j*=1,2 and 3). Thus, *H*_{j}(*k*)=*H*(*k*), where *H*(*k*) is defined by equation (3.16) without the subscript (*j*). By substituting *ρ*_{j}(*k*) with *H*_{j}=*H* in the global relation (2.4) we find(3.20)The contributions from the corner terms, *C*_{j}, cancel. Indeed, this contribution is proportional to the following expression:(3.21)However, the assumption of continuity at the vertices implies(3.22)Hence, the terms *q*^{(1)}(−*l*/2) and *q*^{(2)}(*l*/2) in the expression (3.21) cancel iff(3.23)This equation is indeed valid and is the consequence of the identity

Given that the corner terms cancel, the global relation and its Schwarz conjugate yield the following equations:(3.24)where *A*(*k*) and *B*(*k*) are defined by equations (3.6) in terms of *F*_{j} (compare with equations (1.15) and (1.16)). Let(3.25)We find the following expression for in terms of following precisely the same steps used for the general Dirichlet problem:(3.26)where *T*(*k*) is given in terms of *F*_{j}(*k*) by the following equation:(3.27)If *k*=*k*_{m}, where *k*_{m} is defined bythenThus, evaluating equation (3.26) at *k*=*k*_{m}, we find the following expression:(3.28)where(3.29)Even if one is able to invert equation (3.28) for the bracket appearing in the integrant of the left-hand side of equation (3.28), one will find an expression which will involve an infinite series over the transcendental values of *k*_{m}. Thus, instead of analysing the relevant inversion, we will analyse the general Robin problem using the generalized Fourier transform approach (see §5).

In the case of the Neumann problem, equations (3.28) and (3.29) simplify and yield the following result.

*Let the real valued function q*(*x*,*y*) *satisfy equation* *(1.7)* *in the triangular domain D,* *with the Neumann boundary conditions*(3.30)*where the functions f*_{j}(*s*) *have sufficient smoothness and are continuous at the vertices of the triangle.* *Then,* *the Dirichlet data q*^{(j)}(*s*) (*j*=1,2 *and* 3) *can be expressed in terms of the given Neumann data by the Fourier series*(3.31)*where* *are given by equation* *(3.12)* *and*(3.32)*The known function T*_{N}(*k*_{m}) *is defined by the equation*(3.33)*where F*_{j}(*k*) *is given by equation* *(3.17)* *with β*_{j}=1.

## 4. Poincaré type boundary-value problems

Suppose that *q*(*x*,*y*) satisfies the Poincaré type boundary condition (1.11). Then substituting the expression *ρ*_{j}(*k*) from equation (3.15) into the global relation (2.4), we find an equation similar to equation (3.20), where *H*(*k*), and *H*(α*k*) are replaced by *H*_{1}(*k*), and *H*_{3}(*αk*), and *F*_{j}, *C*_{j} and *Y*_{j} are defined by equations (3.17)–(3.19). Proceeding as in §3*c*, in analogy with equation (3.26), we now find(4.1)where *T*(*k*) is defined in terms of the known functions *f*_{j}(*s*), *C*(*k*) involves the values of *q* at the corners and *D*(*k*) and are defined by the following equations:(4.2)(4.3a)(4.3b)and(4.3c)with(4.4)In order to be able to solve this problem using a generalized Fourier integral we require that when *D*(*k*) vanishes *Γ*_{2}(*k*) and *Γ*_{3}(*k*) are proportional to *Γ*_{1}(*k*). Actually, *Γ*_{3}(*k*) is proportional to *Γ*_{1}(*k*) for all complex *k*, provided that(4.5a)By equating the brackets appearing in the definitions of *Γ*_{1}(*k*) and *Γ*_{2}(*k*) and replacing the resulting expression *e*^{6}(*k*) withit follows that *Γ*_{2}(*k*) is proportional to *Γ*_{1}(*k*) provided that(4.5b)Equation (4.5*b*) is valid if the following two equations are valid:(4.6a)(4.6b)Indeed, in order to simplify equation (4.5*b*) we first compute the product :(4.7)The function can be obtained from *H*_{1}(*k*) by replacing *β*_{1} with −*β*_{1}; thus, is given by an expression similar to equation (4.7) with *β*_{1} replaced by −*β*_{1}. Hence equation (4.5*b*) yieldsThis equation simplifies to the equationwhich is valid for all *k* iff equations (4.6*a*,*b*) are valid.

Equation (4.6*a*) implies then and we find equation (1.12*b*). Similarly, equation (4.5*a*) yields equation (1.12*c*).

### (a) The case where the corner terms cancel

The definition of the corner terms *C*_{j}(*k*), that is, equation (3.18), shows that *C*_{j}(*k*) involves exp[i*β*_{j}]/sin *β*_{j}. Thus, the contribution of the corner terms in the global relation (3.20) vanishes iff(4.8)

(4.9)In this case,(4.10)

(4.11)In this case,(4.12)In particular, if then(4.13)Thus,(4.14)Hence,(4.15a)(4.15b)(4.15c)and(4.15d)If *P*_{1}(*k*) is defined by equation (4.14) with , it can be verified thatHence,(4.16)

### (b) The Laplace equation

In the particular case of the Laplace equation with *γ*_{j}=0, it follows that , that is, *P*_{j} is independent of *k*.

## 5. Analysis of the global relation via Fourier integrals

In this section we restrict *λ* to be non-negative. Slightly more complicated formulae can be derived for *λ*<0.

We first derive equation (1.25). Letting , the definition of yields(5.1)Suppose that *λ*>0. Letting , it follows that if |*k*|∈(0,+∞), then *t*∈(−∞,∞). Thus, by inverting equation (5.1) we findwhere |*k*| (in the argument of *Ψ*) is a function of *t*. Rewriting *t* in terms of |*k*| we findLetting we obtain(5.2)The right-hand side of this equation equals(5.3)Indeed, for the derivation of equation (5.3) we first observe that the function remains invariant under the transformation . Thus, making the change of variables in the right-hand side of equation (5.2) and usingwe obtain expression (5.3). By combining equations (5.2) and (5.3) we obtain(5.4)Using , this equation becomes equation (1.25). If *λ*=0, we set , , and rewrite (1.14*a*) as(5.5)which is inverted to(5.6)By replacing {i*t*} with in equation (5.6), we obtainThis equation, in comparison to equation (1.25), misses a factor of one-half; this is due to the linearity of the relevant transformation in this case.

### (a) The symmetric Dirichlet problem

Solving equation (1.17) for and substituting the resulting expression in equation (1.25) we find(5.7)where *Δ*(*k*) denotes the coefficient of in equation (1.17), that is,The line splits the complex *k*-plane into the two half planesWe observe that(5.8a)(5.8b)Indeed, the exponential of (5.8*a*) involves which, since *l*/2−*s*≥0, the exponential in (5.8*a*) is bound for , that is, ^{−}. Similarly, the exponential of (5.8*b*) involves which, since *l*/2+*s*≥0, is bound for , that is, ^{+}.

We also note that *Ψ*(*k*)/*Δ*(*k*) is bound for all , *k*≠*s*_{n}. Indeed, *Δ*(*k*) is dominated by *e*(*k*) for Re *k*>0, while *Δ*(*k*) is dominated by *e*(−*k*) for Re *k*<0, hence(5.9)Furthermore, *Ψ*(*k*)*e*(−*k*) involves *k*(*s*−*l*/2), which is bounded for Re *k*≥0, while *Ψ*(*k*)*e*(*k*) involves *k*(*s*+*l*/2), which is bounded for Re *k*≤0 (recall that −*l*/2≤*s*≤*l*/2).

The above considerations imply that the parts of the integral (5.7) containing and can be computed using Cauchy's theorem in ^{−} and ^{+}, respectively. The associated residues can be computed as follows: let and denote the subsets of *s*_{n} in ^{+} and ^{−}, respectively. Evaluating equation (1.17) at we findthus(5.10)

### (b) The Poincaré problem

Evaluating equation (4.1) at *k*=*k*_{m}, where *k*_{m} is a zero of *D*(*k*), it follows that the unknown terms *Y*_{j}(*k*) appear in the formThe crucial difference of this general case, as compared with the oblique Robin case (1.9), is the following: using the definition of *Y*_{j}(*k*_{m}) we find that the coefficients of *q*^{(2)}(*s*) and of *q*^{(3)}(*s*) involve, in general, *k*_{m}-dependent expressions, thus it is not clear how the associated integral can be inverted. In contrast, equation (4.1) can be solved using the approach of §5*a*. The definition of , i.e. equation (3.19), and equation (1.25), imply(5.11)Solving equation (4.1) for and substituting the resulting expression in equation (5.11), we find an integral involving the three unknown functions . The unknown part of this integral involves the factors (5.8*a*,*b*) analysed already, as well as factors of the typeThese terms are bounded for all *k*≠*k*_{m}. Indeed, ignoring the terms involving *P*_{j}(*k*) we findwhich are identical with the expressions (5.9) except for the occurrence of the factors *e*^{4}(−*k*) and *e*^{4}(*k*) for Re *k*>0 and Re *k*<0, which are bounded.

The above discussion implies that the integral involving(5.12)can be computed by using Cauchy's theorem in ^{+}. Evaluating equation (4.1) at it follows that the associated residue equals(5.13)Similarly, the integral involving(5.14)can be computed by using Cauchy's theorem in ^{−}. Evaluating equation (4.1) at , it follows that the associated residue equals(5.15)

In what follows, we give the details for a mixed Neumann–Robin problem.

We will consider example 4.2, as it is described by (4.11) with On side (1) we assume the Robin condition(5.16)and on sides (2) and (3) we assume the Neumann conditions(5.17)and(5.18)Then *H*_{j}(*k*), *P*_{j}(*k*), *D*(*k*) and *Γ*_{j}(*k*), are given by (4.13), (4.14), (4.15*a*–*d*), respectively. Furthermore, *Γ*_{3}(*k*) is proportional to *Γ*_{1}(*k*) (see equation (4.16*a*)), while *Γ*_{2}(*k*) becomes proportional to *Γ*_{1}(*k*) only on those *k*_{m} for which *D*(*k*) vanishes. These are roots of the transcendental equation(5.19)For *k*=*k*_{m}, equation (4.1), in view of (4.16), implies(5.20)By virtue of (4.15*b*) and the identity(5.21)equation (5.20) is written as(5.22)Since the corner terms *C*(*k*) vanish, the representation (5.11) and equation (4.1) yield(5.23)Utilising the expression (5.22), we arrive at the following result.

*Let the real valued function q*(*x*,*y*) *satisfy equation* *(1.7)* *with λ*>0 *in the triangular domain D,* *with the Robin boundary condition* *(5.16)* *on side* (1) *and the Neumann boundary conditions* *(5.17)* *and* *(5.18)* *on sides* (2) *and* (3)*, where the given functions f*_{j}(*s*) *have sufficient smoothness and are continuous at the vertices*. *Then the Dirichlet value on side* (2) *is given by*(5.24)*where* *denotes the derivative of* *D*(*k*) *evaluated at* . *The summations are taken over all* *and* *, respectively, and* *T*,*H*_{2},*P*_{1} *are defined by equations* *(3.27), (3.16), (4.4),* *respectively.*

There exist similar formulae for the Dirichlet values on sides (1) and (3).

## 6. The integral representations

If *λ*≥0, the classical Green's representation is given by Dassios & Kleinman (2000),(6.1)where the integration is over the boundary ∂*D* of the triangle in the positive direction, ∂_{n′} denotes the outward normal derivative on ∂*D*, d*l*(* r*′) is the line element along ∂

*D*, and

*K*(

*x*) is the modified Bessel function of the zeroth order and of the second kind for the modified Helmholtz equation. For the case of the Helmholtz equation

*K*(

*x*) is proportional to the Hankel function of the zeroth order and of the first kind, while for the Laplace's equation

*K*(

*x*) is proportional to the logarithm of

*x*.

For the Laplace equation, the integral representation constructed in Fokas (2001) is defined as follows:(6.2)where the contours *l*_{j} are the rays from 0 to ∞ specified by the arguments and respectively, and the functions are defined by equations (2.6) in terms of *ρ*_{j}(*k*), where the latter functions are defined by equations (2.7), (1.14*a*,*b*) with *λ*=0 (figure 2).

For the modified Helmholtz equation, the analogue of equation (6.2) is (Fokas 2001)(6.3)where the rays *l*_{j} are the same as in (6.2) and are defined by equations (2.6) and (2.7).

There exists a similar representation for the Helmholtz equation, which however, in addition to rays, also involves circular arcs (Fokas 2001).

### (a) The symmetric Dirichlet problem

Using the integral representation (6.3) it is possible to compute directly , bypassing the computation of the unknown boundary values. For brevity of presentation we will only give details for the symmetric Dirichlet problem. The analysis of the more general boundary-value problems (1.8)–(1.11) is similar.

Recalling the definitions of , i.e. equations (2.6) and (2.7), it follows that the representation given by equation (6.3) involves the known function *F*(*k*) defined by equation (3.2), as well as the unknown function *Ψ*(*k*), which on the rays *l*_{j} appears as(6.4)Solving equation (1.17) for in terms of *Ψ*(*k*), and then using the Schwarz conjugation of the resulting equations in order to express *Ψ*(α*k*) in terms of *Ψ*(*k*), it follows that the expressions in (6.4) involve the unknown function *Ψ*(*k*)/*Δ*(*k*), *Δ*(*k*)=*e*(*k*)−*e*(−*k*), times the following expressions:(6.5)The third of the relations in (1.5) implies ; thusReplacing *k* by and by *αk* in these identities, we findThus the expressions in (6.5) involveHence, the unknown part of involves the following integrals:(6.6)(6.7a)(6.7b)(6.7c)Each of the above integrals can be computed in terms of residues. Indeed, it was shown in §5 that *Ψ*(*k*)/*k**Δ*(*k*) is bounded as *k*→0 and as *k*→∞. Furthermore, it will be verified below that the exponentials(6.8)are bounded as *k*→0 and *k*→∞, for arg *k* inrespectively, provided that . We first consider the first exponential in (6.8); since , this exponential can be written as . If *z* is in the triangular domain thenThus if we findHence exp{i*k*(*z*−*z*_{2})} is bounded as |*k*|→∞ and is bounded as |*k*|→0. Hence there is no contribution from zero and from infinity.

The results for the second and the third integrals in (6.8) follows from the above result by using appropriate rotations. The roots of *Δ*(*k*)=0 lie on the imaginary axis. Denote by those with a positive imaginary part and by those with a negative imaginary part. Obviously, the residue from each has a full contribution to , while the residue contribution from each is split into two halves, one half is contributed to and one half to .

Tedious, but straightforward, calculations lead to the expression(6.9)In the above calculations, the value of is obtained from (1.18).

We observe that for each (6.10)Thus, this expression shows that the equilateral triangle admits separable solutions. It is clear that each eigensolution in (6.10) solves equation (1.7).

## 7. Conclusion

Eigenvalues and eigenfunctions for equation (1.7) with homogeneous Dirichlet, Neumann, and Robin boundary conditions were constructed in the classical works of Lamé (1833, 1852, 1861). Some of these results have been rederived by several authors, in particular the Dirichlet problem is discussed in a recent review (McCartin 2003). The Robin problem is analysed in Shanin (1997). It is remarkable that Lamé argued, using physical considerations, that it is impossible to solve certain problems using infinite series as opposed to integrals. Indeed Lamé writes: The series should therefore express the fact that the temperature remains zero on strips of constant width separated by other strips of double width, in which the temperature may vary. The analytic interpretation of this sort of discontinuity demands the introduction of terms where the variables appear

*inside integrals*. These terms, of a nature that we will not consider here, cannot disappear from the total series unless the discontinuity disappears.

(Lamé 1861, p. 191; emphasis added)

In this paper we have solved several boundary-value problems by introducing a novel analysis of the global relation, i.e. of equation (1.13). Although this equation was first derived in the important work of Shanin (1997), where it was also used to solve the Robin problem, our treatment of equation (1.13) is different from that of Shanin (1997). As a consequence of our novel analysis of equation (1.13) we are able to first present a straightforward treatment of *simple* boundary-value problems. This treatment, which is based on the evaluation of the basic algebraic relations (see §1) at particular values of *k*, expresses the unknown boundary values in terms of infinite series. The Dirichlet, Neumann and Robin problems can be solved using this approach. We then show that, in agreement with the above remarks of Lamé, more *complicated* boundary-value problems apparently require the use of generalized Fourier integrals as opposed to infinite series. Proposition 5.1 presents the solution of such a problem.

In this paper, as opposed to the works of ben-Avraham and Fokas (1999, 2001), Fokas (2001), Fokas & Kapaev (2003), Antipov & Fokas (2004) and Crowdy & Fokas (2004), we have introduced a method for determining the *generalized Dirichlet to Neumann map*, i.e. determining the *unknown boundary values* as opposed to determing *q*(*x*,*y*) itself. In this respect we note that (i) in some applications one requires precisely these unknown boundary values, and (ii) when both the Dirichlet and the Neumann boundary values are known, it is straightforward to compute *q*(*x*,*y*).

We emphasize, however, that the approach of §5 can be used to construct directly *q*(*x*,*y*). Indeed, if one uses the novel integral representations for *q*(*x*,*y*) obtained in Fokas (2001), instead for the representation (1.25) for *q*_{N}, and if one follows the approach of §5, one can again compute explicitly the contribution of the unknown functions. This latter approach is illustrated in §6*a* for the symmetric Dirichlet problem. More complicated problems using this approach are solved in ben-Avraham and Fokas (1999, 2001), Fokas & Kapaev (2003) and Crowdy & Fokas (2004).

In order to compute *q*(*x*,*y*) from the knowledge of both the Dirichlet and the Neumann boundary values one can use either the classical Green's formulae or the representations of Fokas (2001). Regarding the latter representations, we note that they provide a tailor-made transform for the particular problem at hand. In fact, the exponential reflects the structure of the PDE, the contours *l*_{j} in the complex *k*-plane reflect the geometry of the domain, and the functions *ρ*_{j}(*k*) describe the boundary conditions.

Both the Dirichlet and the Neumann problems involve elementary trigonometric functions. It is interesting that the analysis of the global condition yields these separable solutions without the direct use of separation of variables.

For arbitrary values of the constants *β*_{j} and *γ*_{j}, the Poincaré problem (1.11) gives rise to a matrix Riemann–Hilbert problem. For the particular case that equations (1.12*a*,*b*,*c*) are valid, it is possible to avoid this Riemann–Hilbert problem and to solve the problem in closed form. Although equations (1.12*a–c*) impose severe restrictions on *β*_{j} and *γ*_{j}, some of the resulting cases appear interesting. These cases include the following.

,

*γ*_{j}arbitrary. In this case the angles are specified, but*γ*_{j}are arbitrary.The mixed Neumann–Robin problem analysed in §5.

*γ*_{1}=*γ*_{2}=*γ*_{3}=*γ*.

In this case all derivatives are computed along a direction making an angle *β* with the positive vertical axis.

The results presented here can be made rigorous, following a formalism similar to the one used in Fulton *et al*. (2004).

Several problems remain open which include the following.

The investigation of singularities associated with discontinuous boundary conditions.

If the

*β*_{j}differ, then the global relation (3.20) contains a contribution from*q*at the three corners. Several approaches for determining these terms are presented in Fokas & Kapaev (2003) and Antipov & Fokas (2004); however, the optimal treatment of these terms remains open.

The approach introduced in Fokas (1997, 2001) constructs the solutions of a given boundary-value problem *without* using eigenfunction expansions. Similar considerations apply to the approach introduced here for constructing the generalized Dirichlet to Neumann map. However, it turns out that the above approaches can also be used to investigate the existence of eigenfunction expansions and to construct these expansions when they exist. This will be presented elsewhere.

## Acknowledgments

This work was partially supported by the EPSRC. This is part of a joint programme undertaken with A.C. Newell.

## Footnotes

- Received June 3, 2004.
- Accepted February 7, 2005.

- © 2005 The Royal Society