This can be done numerically in Mathematica. Here is an example.
sol = NDSolve[{Laplacian[u[x, y], {x, y}] == 0, DirichletCondition[u[x, y] == 0, x <= -0.3],
DirichletCondition[u[x, y] == 1, x >= 0.35]}, u, {x, y} \[Element] Disk[]]
ContourPlot[u[x, y] /. sol, {x, -1, 1}, {y, -1, 1}]
