Reflection in R2

Suppose we have two disjoint discs {D_1=B(c_1,r_1)} and {D_2=B(c_2,r_2)} on {\mathbb{R}^2}. We want to find a bounded harmonic function such that it is constant on {\partial D_1} and on {\partial D_2} respectively.

Denote {R_i} is the reflection with respect to {D_i}, {i=1,2}. We have the following important identity for x\in \partial D_i

\displaystyle |x-p|=|x-R_i(p)|\frac{|p-c_i|}{r_i},\quad \text{for any }p

Pick any {p} and define

\displaystyle h_0(x)=\ln \frac{|x-R_1(p)|}{|x-p|}

Then using the identity it is easy to see {h_0(x)=\ln \frac{r_1}{|p-c_1|}} on {\partial D_1}. However, it is not constant on {\partial D_2}. We need to modify it. Using {|x-R_1(p)|=|x-R_2R_1(p)|\frac{|R_1(p)-c_2|}{r_2}} for {x\in\partial D_2}. Define 

\displaystyle h_1(x)=-\ln\frac{|x-R_2R_1(p)|}{|x-p|}

Consider {h_0+h_1}

\displaystyle h_0+h_1=\ln \frac{|x-R_1(p)|}{|x-R_2R_1(p)|}

then it is constant on {\partial D_2}, but not constant on {\partial D_1}. Using {|x-R_2R_1(p)|=|x-R_1R_2R_1(p)|\frac{|R_2R_2(p)-c_1|}{r_1}}. Construct 

\displaystyle h_2=-\ln \frac{|x-R_1(p)|}{|x-R_1R_2R_1(p)|}

Consider 

\displaystyle h_0+h_1+h_2=\ln \frac{|x-R_1R_2R_1(p)|}{|x-R_2R_1(p)|}

then {h_0+h_1+h_2} is constant on {\partial D_1}. Compare {h_0+h_1+h_2} with {h_0}, we can see that {p} is changed to {R_2R_1(p)}. We can continue this process any finite times. Suppose that {(R_2R_1)^kp\rightarrow p_*} as {k\rightarrow \infty}. It is possible to get a desired function. If so, then {p_*} is the fixed point of {R_2R_1}. Actually {R_2R_1} has two fixed points, one is inside {D_1}, denote as {p_1}, and the other one is {R_1p_1}, denote as {p_2}. Consider the following function, it is easy to verify that {h} is constant on {\partial D_1} and on {\partial D_2} respectively.

\displaystyle h=\ln\frac{|x-p_1|}{|x-p_2|}

[1] Lim, M., Yun, K. H. (2009). Blow-up of electric fields between closely spaced spherical perfect conductors. Communications in Partial Differential Equations, 34(10), 1287-1315.

By Sun's World, on March 6, 2024 at 2:10 am, under Elliptic PDE, PDE. Tags: . No Comments
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