h = ImplicitRegion[1 <= x^2 + y^2 <= 10, {x, y}];
sol = NDSolveValue[{[Rho]^2 D[
        u[[Rho], [Phi]], [Rho], [Rho]] + [Rho] D[
        u[[Rho], [Phi]], [Rho]] +
      D[u[[Rho], [Phi]], [Phi], [Phi]] == 0,
    DirichletCondition[
     u[[Rho], [Phi]] == 100., [Rho] == 1 &&
      0 <= [Phi] <= 2 [Pi]],
    DirichletCondition[
     u[[Rho], [Phi]] == 0., [Rho] == 8 && 0 <= y <= 2 [Pi]]},
   u[[Rho], [Phi]], {[Rho], [Phi]} [Element] h];

DensityPlot[sol[x, y], {x, 0, 100}, {y, 0, 100}, Mesh -> None,
 ColorFunction -> "Rainbow", PlotLegends -> Automatic]