Fundamental Solution of Elliptic Equation with Positive Definite Matrix Coefficient

with coefficient defined in an n −dimensional domain D, where bi(x) and c(x) are any functions that depend on x. L is said to be of elliptic type (or elliptic) at a point x if the matrix aij(x 0) is positive definite, i.e., if for any real vector ξ ≠ 0, ∑ ∑ aijξiξj > 0 n j=1 n i=1 [3]. Laplace's equation ∆u = 0 is the simplest and most basic example of elliptic equation, and the Laplacian of u is ∆u = ∑ uxixi n i=1 [1]. While, a symmetric matrix A is called a positive definite matrix if a quadratic form x Ax > 0 for all x ≠ 0, x ∈ R [5]. A fundamental solution of the differential operator L in Ω is a function K(x, z) defined for xεΩ, zεΩ, x ≠ z and satisfying the following property: For any bounded domain R with smooth boundary ∂R and for any zεR,


INTRODUCTION
We consider the linear differential operator  = ∑ ∑   ()  2      + ∑   () with coefficient defined in an  −dimensional domain , where   () and () are any functions that depend on . is said to be of elliptic type (or elliptic) at a point  0 if the matrix   ( 0 ) is positive definite, i.e., if for any real vector  ≠ 0, ∑ ∑       > 0  =1  =1 [3].Laplace's equation ∆ = 0 is the simplest and most basic example of elliptic equation, and the Laplacian of  is ∆ = ∑       =1 [1].While, a symmetric matrix  is called a positive definite matrix if a quadratic form    > 0 for all  ≠ 0,  ∈ ℝ [5].
A fundamental solution of the differential operator  in Ω is a function (, ) defined for Ω, Ω,  ≠  and satisfying the following property: For any bounded domain ℝ with smooth boundary ℝ and for any ℝ, () = ∫ (, ) * () ̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅ , ℝ for every  ∈  0  (ℝ), and  * is formal adjoint of  [4].Here, we define the following equation as elliptic equation with positive definite matrix ∑ ∑     (  ) = 0,  =1  =1 (2) where   is element of positive definite matrix,  = () and  ∈ ,  ⊂ ℝ  .In 2016, Fitri studied the Holder regularity of weak solutions to linear elliptic partial differential equations with continuous coefficients, Campanato type estimates are obtained for the validity of regularity of solutions (see [2] ).In this paper, we study the fundamental solution of elliptic equation with positive definite matrix coefficient.Due to the symmetry of elliptic equation, radial solutions are natural to look for since the given partial differential equation can be reduced to an ordinary differential equation which is easier to solve.In this way, we can reduce the higher dimensional problems to one dimensional problems.

Fundamental Solution of Elliptic Equation with Positive Definite Matrix Coefficient
Khoirunisa 65

RESULTS AND DISCUSSION
Let the elliptic equation with positive definite matrix as in (2) then to find a solutions  of elliptic equations , it consequently seems advisable to search for radial solutions, that is functions of .where , ,  ∈ ℝ.

Remarks:
We suppose the identity matrix  × as a coefficient of (2). is a positive definite matrix, then the elliptic equation can be formed as follows .

∑
∑     (  ) ≠ .According to   and   as above, equation (7) can be written as∑   (  ) = 0,  =1where (8) is equivalent to the Laplace's equation ∑       =1 where   is element of coffactor matrix   .Then partial derivative of r with respect to xj and partial derivative of r with respect to xixj are defined by  with respect to     denoted   (  ) =