Green's Functions of Partial Differential Equations with Involutions

In this paper we develop a way of obtaining Green's functions for partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat transfer in a conducting plate which is bent in half.


Introduction
The study of differential equations with involutions dates back to the work of Silberstein [10] who, in 1940, obtained the solution of the equation f (x) = f (1/x). In the field of differential equations there has been quite a number of publications (see for instance the monograph on the subject of reducible differential equations of Wiener [11]) but most of them relate to ordinary differential equations (ODEs). There has also been some work in partial differential equations (PDEs), for instance [11] or [2], where they study a PDE with reflection.
In what Green's functions for equations with involutions is concerned, we find in [3] the first Green's function for ODEs with reflection and in [4] we have a framework that allows the reduction of any differential equation with reflection and constant coefficients. This setting is established in a general way, so it can be used as well for other operators (the Hilbert transform, for instance) or in other yet unexplored problems, like PDEs [8]. In this work we take this last approach and find a way of reducing general linear PDEs with linear involutions to usual PDEs.
The paper is structured as follows. In Section 2 we develop an abstract framework, with definitions and adequate notation in order to treat linear PDEs as elements of a vector space † Partially supported by Xunta de Galicia (Spain), project EM2014/032. ‡ Partially supported by Spanish MICINN Grant with FEDER funds MTM2014-52232-P.
consisting of symmetric tensors. This will allow us to systematize the algebraic transformations necessary in order to obtain the desired reduction of the problem. In Section 3 we start providing a simple example that shows how the general process works and then prove the main result of the paper, Theorem 3.3, that permits a general reduction in the case of order two involutions. We end the Section with a problem with an order 3 involution (Example 3.4), illustrating that the same principles could be applied to higher order involutions. Finally, in Section 4, we describe a way to obtain Green's functions for PDEs with linear involutions and apply it to a model of the process of heat transfer in a conducting plate which is bent in half with the two halves separated by some insulating material. We study the problem for different kinds of boundary conditions and a general heat source.

Derivatives
Let be or , n ∈ and Ω ⊂ V := n a connected open subset. For p ≥ 2, note by V p the space of symmetric tensors or order p, that is, the space of tensors of order p modulus the permutations of their components. We note V 1 = V and V 0 = . For the convenience of the reader, we summarize now the properties and operations of the symmetric tensors: With these properties, V p is an -vector space of dimension For every v = (v 1 ,..., v n ) ∈ V , we define the directional derivative operator as If ∇ y denotes the gradient vector of y, then D v ( y) = v T ∇ y. Observe that D λu+v = λD u + D v for every u, v ∈ n and λ ∈ , that is, where v u denotes de symmetric tensor product of u and v. In the same way, we define the composition of higher order derivatives by D p where ω 1 ∈ V q and ω 2 ∈ V p , p,q ∈ .
In this way, a linear partial differential equation is given by where ω k ∈ V k for k = 1,..., m and D 0 ω 0 u ≡ ω 0 u where ω 0 ∈ (that is, V 0 := ). Now, the operator L can be identified with ω 0 + ω 1 + ··· + ω n , which is an element of the symmetric tensor algebra It is interesting to point out the the Hilbert space completion of S * V , that is, F + (V ) := S * V , is called the symmetric or bosonic Fock space, which is widely used in quantum mechanics [5].

Involutions dfn 2.1.
Let Ω be a set and A : Ω → Ω, p ∈ , p ≥ 2. We say that A is an order p involution if We will consider linear involutions in n . They are characterized by the following theorem.

thm 2.2 ([1]). A necessary and sufficient condition for a linear transformation A on a finite dimensional complex vector space V to be an involution of order p is that
is a p-th root of the unity, and P 1 ,..., P k are projections such that P j P l = 0, i = j and P 1 + ··· + P k = Id. rem 2.3. As an straightforward consequence of this result we have that there are only order two linear involutions in n . This is because the only real p-th roots of the unity are contained in {±1}.
The characterization provided in Theorem 2.2 can be rewritten in the following way. Proof. Consider the characterization of involutions given by Theorem 2.2. Take the vector subspaces H j := P j V , j = 1,..., k. Then, V = H 1 ⊕ ··· ⊕ H k . Take U −1 to be the matrix of which its columns are, consecutively, a basis of H k . Hence,

cor 2.4. A necessary and sufficient condition for a linear transformation A on V to be an involution of order p is that
where every α j is repeated accorollaryding to the dimension of H k .

Pullbacks and equations
Let (Ω, ) be the set of functions from Ω ⊂ n to . We define the pullback operator by a function ϕ ∈ (Ω,Ω) as Assume A is a linear order p involution on Ω (Ω has to be such that Ω = A(Ω)). From now on, we will omit the composition signs. Observe that, for v ∈ V , x ∈ Ω and y ∈ 1 (Ω, ), or, written briefly, Aω k . We can consider now linear partial differential equations with linear involutions of the form where ω j k ∈ V k for k = 0,..., m; j = 0,..., p − 1. This time we can identify L with The interest in these equations appears when they can be reduced to usual partial differential equations.

dfn 2.5 ([4]). If [D]
is the ring of polynomials on the usual differential operator D and is any operator algebra containing [D], then an equation L x = 0, where L ∈ , is said to be a reducible differential equation if there exits R ∈ such that RL ∈ [D].
In our present case, the first projection of the algebra (S * V ) p is precisely the algebra of partial differential operators on n variables PD n [ ], so we want to find elements R ∈ (S * V ) p such that they nullify the last p − 1 components of L.

Reducing the operators
We start with an illustrative example.
A is an order 2 involution. Consider the equation Here we work with the operator L = D v + A * . Take then R = D −Av + A * and consider the identity operator Id. We have that Hence, every two-times differentiable solution of equation (3.1) has to be a solution of the partial differential equation rem 3.2. With the notation we have introduced, it is extremely important the use of parentheses.
So it is enough to check that, for s = 0,...,2m, Substituting the ξ j k by their given values, Let us see that L and R commute.
On the other hand, Hence, the result is proven.
Similar reductions can be found for higher order involutions, although the coefficients may have a much more complex expression. EXA 3.4. Let A be and order 3 linear involution in n , v ∈ n \{0} and consider the operator Observe that second derivatives occur in R but not in L. We have that Unfortunately, we do not have commutativity in general: In the particular case v is a fixed point of A, RL = LR.
The obtaining of a general expression for associated operators in the case of order 3 involutions and the conditions under which such operators commute is an interesting open problem.
Let R ∈ (S * V ) p , f ∈ L 1 ( n , ) and consider the problem Given a function G : n × n → , we define the operator H G such that H G (h)| x := n G(x,s)h(s)ds for every h ∈ L 1 ( n , ), assuming such an integral is well defined. Also, given an operator R for functions of one variable, define the operator R as R G(t,s) := R(G(·,s))| t for every s, that is, the operator acts on G as a function of its first variable.
We have now the following theorem relating problems (4.1) and (4.2). The proof for the case of ordinary differential equations can be found in [4]. The case of PDEs is analogous.
. Assume L commutes with R and that there exists G such that H G is well defined satisfying Then, v := H G f is a solution of problem (4.2) and u := H R G h is a solution of problem (4.1).

A model of stationary heat transfer in a bent plate
We now consider a circular plate which is bent in half, with each of the two distinct halves separated by a very small distance which may be filled with some kind of (imperfect) heat insulating material (see Figure 4.1).
The heat equation which determines the temperature u on the plate for this situation is given by is the usual heat equation with heat transfer coefficient α > 0 and the term that goes with β > 0 relates to the heat transfer from the corollaryresponding point in the other half of the plate due to Newton's law of cooling.
In general, the functions with µ, ν ∈ , are Green's functions related to the operator ∆ 2 with different boundary conditions. The associated function for the operator L is given by G 6 (η,ξ) = 1 + µ + log η − ξ 2π .