$$U_{xx}+U_{yy}=0$$
with $U = 1$ on $x^2+y^2=1$ and $\partial U/\partial n = 1$ on $x^2+y^2=1.$ ($n$=normal).
So I tried switching $U_{xx}+U_{yy}=0$ to polars and got $U_{rr}+(1/r^2)U_{\theta \theta} + (1/r)U_r=0$. But not sure what to do next. Any suggestions on this.