Math 6420 • PDE

Solve Laplace's equation
\[
  u_{xx}+u_{yy}=0
  \quad\text{in the domain}\quad
  D=\{(x,y)\in\mathbb{R}^2:\ (x-1)^2+y^2>1\ \&\ (x-2)^2+y^2<4\}
\]
subject to the boundary condition
\[
  u(x,y)=a\text{ when }(x-1)^2+y^2=1
  \qquad\text{and}\qquad
  u(x,y)=b\text{ when }(x-2)^2+y^2=4
\]
[\(a\) and \(b\) are real parameters; the two circles touch each other; express your answer in
terms of the original real variables \(x,y\)].