The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients : The Simple Layer Potential Ansatz

and Applied Analysis 3 Proof. Taking definition (13) into account, we have


Introduction
As remarked in [1, p. 121], elliptic operators with variable coefficients naturally arise in several areas of physics and engineering.In this paper, we study the Dirichlet problem related to a scalar elliptic second-order differential operator with smooth coefficients in divergence form in a bounded simply connected domain of R  ( ≥ 3) with Lyapunov boundary.This is a classical problem which nowadays can be treated in several ways.In particular, different potential methods have been developed for such operators (see, e.g., [1][2][3][4][5][6]).
In the present paper, we obtain the solution of the Dirichlet problem by means of a simple layer potential instead of the classical double layer potential (see, e.g., [6, pp. 73-75]).We use an indirect boundary integral method introduced for the first time in [7] for the -dimensional Laplacian.It requires neither the knowledge of pseudodifferential operators nor the use of hypersingular integrals, but it hinges on the theory of singular integral operators and the theory of differential forms (for details of the method, see, e.g., [8, Section 2]).The method has been also used to treat different boundary value problems in simply connected domains: the Neumann problem for Laplace equation (via a double layer potential), the Dirichlet problem for the Lamé and Stokes systems, the four boundary value problems of the theory of thermoelastic pseudooscillations, the traction problem for Lamé and Stokes systems, the four basic boundary value problems arising in couple-stress elasticity, and the two boundary value problems of the linear theory of viscoelastic materials with voids (see [9,10] and the references therein).The method can be applied also in multiply connected domains, as shown for the Laplacian, the linearized elastostatics, and the Stokes system (see [11] and the references therein).
The present paper is organized as follows.
In Section 2, after giving preliminary results, we make use of Fichera's construction of a principal fundamental solution [12] and we prove some identities for the related nuclear double form.
Section 3 is devoted to the study of the Dirichlet problem.It contains the main results concerning the reduction of a certain singular integral operator acting in spaces of differential forms and the integral representation of the solution of the Dirichlet problem by means of a simple layer potential.

Preliminary Results
Let Ω be a bounded domain (open connected set) of R  ( ≥ 3).
We suppose that the coefficients   are defined on ,  being an open ball containing Ω, and we assume that they belong to  2, (), 0 <  ≤ 1.
For the sake of simplicity, we suppose that the determinant || of  is equal to 1.
A differential form of degree  (in short a -form) on  is a function defined on  whose values are in the -covectors space of R  .A -form  can be represented as with respect to an admissible coordinate system ( 1 , . . .,   ), where   1 ⋅⋅⋅  are the components of a skew-symmetric covariant tensor (for details about differential forms, we refer to [13,14]).The symbol  ℎ  () means the space of all -forms whose components are continuously differentiable up to the order ℎ in a coordinate system of class  ℎ+1 (and then in every coordinate system of class  ℎ+1 ).
If  ∈  1  (), we define the codifferential of  as the following ( − 1)-form: A differential double form  ℎ, (, ) of degree ℎ with respect to  and of degree  with respect to  (in short a double (ℎ, )form) is represented as If ℎ = , we denote it briefly by   (, ).
The next results provide other properties of  and   .

Abstract and Applied Analysis 3
Proof.Taking definition (13) into account, we have On the other hand, and this yields the claim.

Now we pass to show (21). With calculations analogue to (26), we have that
where Arguing again as in (26) and taking Lemma 1 into account, we get where both   −−1, (, ) and   −−1, (, ) are O(|−| 2− ).Then, we obtain the claim by setting Finally, we prove (22).Thanks to ( 9) and ( 19), we have where   ,+1 (, ) = O(|−| 2− ).Now, by using ( 21) and ( 8), we get and hence the claim with where  is a linear first-order differential operator whose coefficients depend only on first-and second-order derivatives of entries of the tensor .
In particular, Proof.We begin by observing that Since  is symmetric and || = 1, we get and, keeping in mind (7), we get On the other hand, and this proves (35).Finally, (36) follows from (35).

The Dirichlet Problem
In this section, we suppose that the domain Ω is such that R  \ Ω is connected and such that its boundary Σ = Ω is a Lyapunov surface (i.e., Σ ∈  1, , 0 <  ≤ 1).
By    (Σ), we denote the space of all -forms whose components are   real-valued functions in a coordinate system of class  1 (and then in every coordinate system of class  1 ).
We will look for the solution of the Dirichlet problem for the operator  in the domain Ω in the form of a simple layer potential.To this end, we introduce the space S  .Definition 5.The function  belongs to S  if and only if there exists  ∈   (Σ) such that it can be represented by means of a simple layer potential; that is, Specifically our aim is to give an existence and uniqueness theorem for the Dirichlet problem First, we prove the following formula.

Proposition 6. For any
where  is the linear first-order differential operator considered in Proposition 3 and Proof.Set, for every  ∉ Σ, On account of ( 22) and (36), we get and (52) follows from (23).
On the other hand, if  ȷ ı is the minor of  −1 obtained deleting the th row and the th column, for  ∈ Ω,  ∈ Σ we get Therefore, Then, if  ∈ Σ, lim and this concludes the proof.
Remark 7. We note that (51) generalizes the following identity (see [7] where (, ) and   (, ) denote the fundamental solution for Laplace equation and the double -form associated with (, ), respectively.
We recall that if  and B are two Banach spaces and  :  → B is a continuous linear operator, we say that  can be reduced on the left if there exists a continuous linear operator   : B →  such that    =  + , where  stands for the identity operator on  and  :  →  is compact.One of the main properties of such operators is that equation  =  has a solution if and only if ⟨, ⟩ = 0 for any  such that  *  = 0,  * being the adjoint of  (see [17,18]).
and then The operator    is compact because of (44).Concerning   , keeping in mind Proposition 6 and setting () = ∫ Σ ()(, )d  , we get From Theorem 8, it follows that operator J can be reduced on the left; therefore, (67) admits a solution  ∈   (Σ) if and only if On the other hand, J *  = 0 if and only if  is a weakly closed form; that is, In fact, if we have and then for any smooth solution  of  = 0 in Ω.Therefore, we have Let us consider If V ∈  ∞ () and  ∈  1 (Ω) ∩  2 (Ω) are such that  = V in Ω and  = 0 on Σ, we have From the Green formulas we have ( The dimension of A is 1. Proof.The Fredholm equation (83) has the same number of linearly independent solutions of the following equation: (88) By Green's formula,  = 0 in  \ Ω and therefore  = 0 in .This implies  1  1 +  2  2 = 0, which is a contradiction.