It seems to work this way:
c1 = Ball[];
c2 = Cone[{{0, 0, 0}, {0, 0, 1/2}}, 1/4];
\[CapitalOmega] = RegionDifference[c1, c2];
op = Laplacian[V[x, y, z], {x, y, z}] == 0;
\[CapitalGamma] = {DirichletCondition[V[x, y, z] == 10,
x^2 + y^2 + z^2 > 9/10],
DirichletCondition[V[x, y, z] == 2, x^2 + y^2 + z^2 < 9/10]};
sol = NDSolveValue[{op, \[CapitalGamma]},
V, {x, y, z} \[Element] \[CapitalOmega]]
