GREEN’S FUNCTION FOR LAPLACIAN

GREEN S FUNCTION FOR LAPLACIANThe Green s FUNCTION is a tool to solve non-homogeneous linear equations. We will illus-trate this idea for the LAPLACIAN .Suppose we want to find the solutionuof the Poisson equation in a domainD Rn: u(x) =f(x),x Dsubject to some homogeneous boundary condition. Imaginefis the heat source anduis thetemperature. The idea of Green s FUNCTION is that if we know the temperature respondingto an impulsive heat source at any pointx0 D, then we can just sum up the result withthe weight functionf(x0) (it specifies the strength of the heat at pointx0) to obtain thetemperature responding to the heat sourcef(x) inD. Mathematically, one may expressthis idea by defining the Green s FUNCTION as the following:Letu=u(x),x= (x1, x2, .. xn) be the solution of the following problem:{ u(x) =f(x),x Dusatisfies some homogeneous boundary condition along the boundary D( )Define the Green s functionG=G(x,x0) to be the solution of{ G(x,x0) = (x x0),x DG(x,x0) satisfies the same homogeneous boundary condition as in ( )( )herex0 Dis a fixed point.}}

In other wards, v should be a solution of the Laplace equation in D satisfying a non-homogeneous boundary condition that nullifies the effect of Γ on the boundary of D. Sim-ilarly we can construct the Green’s function with Neumann BC by setting G(x,x0) = 0)+v(x,x0) where v is a solution of the Laplace equation with a Neumann bound-

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