gg[x_] := 1 - x^2/2
region = ImplicitRegion[x > -1 && x < 1 && y < gg[x] && y > -gg[x], {x, y}];
NDEigensystem[{-Laplacian[{ux[x, y], uy[x, y]}, {x, y}],
DirichletCondition[uy[x, y] == 0, x == -1],
  DirichletCondition[uy[x, y] == 0, x == 1],
  DirichletCondition[{ux[x, y], uy[x, y]}.{1, D[gg[x], x]} == 0,
   y > 0 && x > -1 && x < 1],
  DirichletCondition[{ux[x, y], uy[x, y]}.{1, -D[gg[x], x]} == 0,
  y > 0 && x > -1 && x < 1]}, {ux[x, y],
 uy[x, y]}, {x, y} [Element] region, 6]

region = ImplicitRegion[x^2 + y^2 + z^2 <= 1, {x, y, z}];
g1 = (x* Ex[x, y, z] + y *Ey[x, y, z] + z* Ez[x, y, z])/(x^2 + y^2 + z^2);
NDEigensystem[{-Laplacian[{Ex[x, y, z], Ey[x, y, z], Ez[x, y, z]}, {x,
  y, z}], DirichletCondition[{Ex[x, y, z] - x*g1 == 0,
Ey[x, y, z] - y*g1 == 0, Ez[x, y, z] - z*g1 == 0}, True]}, {Ex[x,  y, z], Ey[x, y, z], Ez[x, y, z]}, {x, y, z} [Element] region, 6]