Fundamental Solution of  Elliptic Equation with Positive Definite Matrix Coefficient


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 5(2) (2018), Pages 64-66 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: 17 January 2018 Reviewed: 09 April 2018 Accepted: 16 May 2018 
DOI: http://dx.doi.org/10.18860/ca.v5i2.4717 
 

Fundamental Solution of  Elliptic Equation with Positive Definite Matrix Coefficient 

Khoirunisa1, Corina Karim2 

1,2Department of Mathematics, Universitas Brawijaya  

 
Email: khoir.n97@gmail.com, co_mathub@ub.ac.id 

ABSTRACT  

We study the fundamental solution of elliptic equations with real constant coefficients   

∑∑𝑎𝑖𝑗𝐷𝑖(𝐷𝑗𝑢)

𝑛

𝑗=1

𝑛

𝑖=1

= 0, 

where 𝑎𝑖𝑗 is a positive definite matrix. We obtained by searching the radial solution such that we solved the 

equation into ordinary differential equations. 
 
Keywords: fundamental solution, elliptic equation, positive definite matrix 

INTRODUCTION  

We consider the linear differential operator  

 𝐿𝑢 = ∑ ∑ 𝑎𝑖𝑗(𝑥)
𝜕2𝑢

𝜕𝑥𝑖𝜕𝑥𝑗
+ ∑ 𝑏𝑖(𝑥)

𝜕𝑢

𝜕𝑥𝑖
+ 𝑐(𝑥)𝑢𝑛𝑖=1

𝑛
𝑗=1

𝑛
𝑖=1 ,             (1) 

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 𝑥0 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 

𝑥𝑇 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, 

𝑢(𝑧) = ∫ 𝐾(𝑥,𝑧)𝐿∗𝑢(𝑥)̅̅ ̅̅ ̅̅̅̅ ̅̅ ̅̅ ̅̅ ̅𝑑𝑥,
ℝ

 

for every 𝑢 ∈ 𝐶0
𝑚(ℝ),  and 𝐿∗ is formal adjoint of  𝐿 [4]. 

Here, we define the following equation as elliptic equation with positive definite matrix  
∑ ∑ 𝑎𝑖𝑗𝐷𝑖(𝐷𝑗𝑢) = 0,

𝑛
𝑗=1

𝑛
𝑖=1                (2) 

where 𝑎𝑖𝑗 is element of positive definite matrix, 𝑢 = 𝑢(𝑥)  and 𝑥 ∈ 𝑈, 𝑈 ⊂ ℝ
𝑛. 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. 

 

mailto:khoir.n97@gmail.com
mailto:co_mathub@


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 𝑢 of elliptic 
equations , it consequently seems advisable to search for radial solutions, that is functions of 𝑟. 
Theorem 3.1 Let 

𝑟 = √∑∑𝑏𝑖𝑗𝑥𝑖𝑥𝑗

𝑛

𝑗=1

𝑛

𝑖=1

, 

where 𝑏𝑖𝑗 is element of coffactor matrix 𝑎𝑖𝑗. Then partial derivative of r with respect to xj and 

partial derivative of r with respect to xixj are defined by 

𝜕𝑟

𝜕𝑥𝑗
=
1

𝑟
∑∑𝑏𝑖𝑗𝑥𝑖

𝑛

𝑗=1

𝑛

𝑖=1

, 

and 

𝜕2𝑢

𝜕𝑥𝑖𝜕𝑥𝑗
=
1

𝑟
∑∑𝑏𝑖𝑗 −

1

𝑟3

𝑛

𝑗=1

𝑛

𝑖=1

∑∑(∑ 𝑏𝑖𝑘𝑥𝑘

𝑛

𝑘=1

∑ 𝑏𝑗𝑘𝑥𝑘

𝑛

𝑘=1

)

𝑛

𝑗=1

𝑛

𝑖=1

. 

 
Theorem 3.2 Suppose  

𝑢 (𝑥) =  𝑣 (𝑟) , 

then the second derivative of  𝑢 with respect to 𝑥𝑖𝑥𝑗 denoted 𝐷𝑖(𝐷𝑗𝑢) =
𝜕2𝑢

𝜕𝑥𝑖𝜕𝑥𝑗
  is the total 

derivative of  𝑣 (𝑟)  respect to 𝑥𝑖𝑥𝑗. 

 

Corollary 3.3 If the second derivative of (4) is substituted into the equation (2)  then we get 

𝑣′′(𝑟)

𝑟2
∑∑𝑎𝑖𝑗 (∑ 𝑏𝑖𝑘𝑥𝑘

𝑛

𝑘=1

∑ 𝑏𝑗𝑘𝑥𝑘

𝑛

𝑘=1

) +
𝑣′(𝑟)

𝑟

𝑛

𝑗=1

𝑛

𝑖=1

∑∑𝑎𝑖𝑗𝑏𝑖𝑗

𝑛

𝑗=1

𝑛

𝑖=1

−
𝑣′(𝑟)

𝑟3
∑∑𝑎𝑖𝑗 (∑ 𝑏𝑖𝑘𝑥𝑘

𝑛

𝑘=1

∑ 𝑏𝑗𝑘𝑥𝑘

𝑛

𝑘=1

)

𝑛

𝑗=1

𝑛

𝑖=1

= 0. 

 
Corollary 3.4 If 𝐴 is positive definite matrix with 𝑎𝑖𝑗 element, and 𝐵 is cofactor matrix of 𝐴 with 

𝑏𝑖𝑗 element then 

𝐼(det𝐴) = 𝐴𝐵. 
 
Corollary 3.5 If 𝐴 matrix and  𝐵 matrix are symmetric, then ∑ ∑ 𝑎𝑖𝑗𝑏𝑖𝑗

𝑛
𝑗=1

𝑛
𝑖=1  can be simplified to 

be 

∑∑𝑎𝑖𝑗𝑏𝑖𝑗 = 𝑛det𝐴.

𝑛

𝑗=1

𝑛

𝑖=1

 

 
Next, to simplify the sum of bracket in the first and the last terms in (5), we denoted 𝑌𝑖 for 𝑖 =
1,2,…,𝑛 and 𝑗 = 1,2,…,𝑛, so that we have 

𝑌𝑖 = (det𝐴)∑∑𝑏𝑖𝑗𝑥𝑖𝑥𝑗

𝑛

𝑗=1

𝑛

𝑖=1

= (det𝐴)𝑟2, 

thus, the equation (5) can be rewrite as 
𝑣′′

𝑣′
=
1 − 𝑛

𝑟
, 

then we have 

(3) 

(5) 

(4) 



Fundamental Solution of  Elliptic Equation with Positive Definite Matrix Coefficient 

Khoirunisa 66 

𝑣′ = 𝑎𝑟1−𝑛. 
 
Since, the domain 𝑥 is at 𝑛-dimension,  𝑛 ≥  2 then  𝑣(𝑟) has the following general form:  
 

𝑣(𝑟) = {
𝑎log𝑟 + 𝑏, 𝑛 = 2
𝑐

𝑟𝑛−2
+ 𝑏, 𝑛 ≥ 3.

 

𝑎,𝑏,𝑐 ∈ ℝ. 

By rescaling (6) to 𝑢(𝑥) = 𝑣(𝑟), where 𝑟 = (∑ ∑ 𝑏𝑖𝑗𝑥𝑖𝑥𝑗
𝑛
𝑗=1

𝑛
𝑖=1 )

1

2, then we get the fundamental 

solution of (2) is 

𝑣(𝑟) =

{
 
 
 

 
 
 
𝑎log(∑∑𝑏𝑖𝑗𝑥𝑖𝑥𝑗

𝑛

𝑗=1

𝑛

𝑖=1

)

1

2

+ 𝑏, 𝑛 = 2

𝑐 (∑∑𝑏𝑖𝑗𝑥𝑖𝑥𝑗

𝑛

𝑗=1

𝑛

𝑖=1

)

2−𝑛

2

+ 𝑏, 𝑛 ≥ 3

 

where 𝑎,𝑏,𝑐 ∈ ℝ. 
 
 
Remarks: 
We suppose the identity matrix 𝐼𝑛×𝑛 as a coefficient of (2). 𝐼 is a positive definite matrix, then the 
elliptic equation can be formed as follows 

∑∑𝑎𝑖𝑗𝐷𝑖(𝐷𝑗𝑢) = 0,

𝑛

𝑗=1

𝑛

𝑖=1

 

where 

𝑎𝑖𝑗 = {
1,   𝑖 = 𝑗
0,   𝑖 ≠ 𝑗.

 

Then coffactor matrix 

𝑏𝑖𝑗 = {
1,   𝑖 = 𝑗
0,   𝑖 ≠ 𝑗.

 

According to 𝑎𝑖𝑗 and 𝑏𝑖𝑗 as above, equation (7) can be written as 

∑𝐷𝑖(𝐷𝑖𝑢) = 0,

𝑛

𝑖=1

 

where (8) is equivalent to the Laplace’s equation ∑ 𝑢𝑥𝑖𝑥𝑖
𝑛
𝑖=1 . 

 

REFERENCES 

[1] Evans, L. C. (1998). Partial Differential Equation. Second Edition. American Mathematical     
Society: USA 

[2] Fitri, S. (2016). Campanato Type Estimates for Solutions of an Elliptic Systems Class. 
International Interdisciplinary Studies Seminar Proceeding. 12-17. Malang: Universitas 
Brawijaya. 

[3]   Friedman, A. (1964). Partial Differential Equation of Parbolic Type. Dover Publication,Inc: 
New York. 

[4]   Friedman, A. (1969). Partial Differential Equation. Dover Publication,Inc: New York. 
[5]   Leon, S. J. (2001). Aljabar Linear dan Aplikasinya. Erlangga: Jakarta. 

 

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