Homogenization for fully nonlinear parabolic equations Original Research Article

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