Symmetrization in parabolic neumann problems

Abstract

We consider the Cauchy-Neumann problem for parabolic operators of the kind:

on a smooth cylinder [0,T]×Ω. By symmetrization techniques we establish for the solution u of this problem an estimate of the kind:

where U is the solution of a symmetrized problem and u(t)*(·) is the decreasing rearrangement of u(t,.). We also obtain an “energy inequality” comparing norms of the gradients of u and U

Keywords:

• Parabolic equations
• rearrangements

AMS(MOS):

• 35K20