  We prove a comparison theorem for the averages of the solutions of two
exterior parabolic problems, the second being the "symmetrization" of the first
one, by using approximation of the Schwarz symmetrization by polarizations, as
it was introduced in [4]. This comparison provides an alternative proof, based
on PDEs, of the isoperimetric inequality for the Wiener sausage, which was
proved in [14].
