I need to find a Green function that provides the value of U at any point within a circle from the values of U at the limit of the circle, knowing that the function U satisfies the two-dimensional Laplace equation at each point in a circle.
I must first express a harmonic function in terms of its limit values, then analyze whether the found expression continues to represent a harmonic function when the limit values are any given continuous function.
I start from the equation derived from Green's third identity, which shows that the value of a harmonic function in R can be found at any point within R strictly from the behavior of U and its normal derivative at the limit of R:
$U(P)=-\frac{1}{4π}\int\int_S(U\frac{∂}{∂n}\frac{1}{r}-\frac{∂U}{∂n}\frac{1}{r})dS$ (1)
Then i must try to eliminate the normal derivative, to simplify the equation, for this purpose i can take the following relationship from Green's second identity, leaving U and V to be harmonics in R:
$$-\frac{1}{4π}\int\int_S(U\frac{∂V}{∂v}-\frac{∂U}{∂v}VdS=0$$
Adding this condition to equation (1) then:
$U(P)=-\frac{1}{4π} \int\int_S(U\frac{∂}{∂n}(V+\frac{1}{r})-(V+\frac{1}{r})\frac{∂U}{∂n}dS$
if i select a harmonic potential V such that $V+\frac{1}{r}=0$ at each point of S, then:
$U(P)=-\frac{1}{4π}\int\int_S (U\frac{∂}{∂n}(V+\frac{1}{r})dS$
Therefore, if for a particular geometry i can find a function V such that: 1- V is harmonic in the entire region R and; 2- $V+\frac{1}{r}=0$ at each point of S
So U can be found throughout the region, and only U values will be required at the limit. The function $V+\frac{1}{r}=0$ is the Green function being the Laplace equation in restricted regions
I consider any regular region R, limited or infinite, and let P (x, y) be any interior point. I take for V, in Green's identity the function $V=\frac{1}{r}$ Where r is the distance from P to Q (ξ, η) where (ξ, η) I must take them as the integration variables in that identity instead of x, y. I take P to be inside R, the identity cannot be applied to the entire region R, so I surround P with a small sigma sphere with P as the center, and remove from R the interior of the sphere.
but I don't know how to continue