Homogenization for fully nonlinear parabolic equations

This paper concerns the homogenization of fully nonlinear parabolic equations of the form∂tuε+H(t,x,t/ε2,x/ε,D2uε)=0in(0,T)×Rn,where the Hamiltonian H(t,x,τ,ξ,X) is periodic both in τ and in ξ. Our aim is to establish sufficient conditions for the convergence (as ε→0) of uε to a solution u to the effective equation∂tu+H¯(t,x,D2u)=0in(0,T)×Rn,where the effective Hamiltonian H¯ is obtained by a parabolic equation called cell problem. We shall prove that H¯ inherits several properties of H. We also consider the case that: uε(0,x)=h(x,x/ε) on Rn; we point out a sufficient condition for having u(0,x)=h¯(x) on Rn, with an effective initial datum h¯ given by the asymptotic behaviour of the solution to the recession problem (a parabolic Cauchy problem related to (1.1)).