一类非线性抛物方程的均匀化

Homogenization of a Class of Nonlinear Parabolic Equations of Second Order

讨论一类非线性抛物方程tuε-d iv(aε(x,uε))=fε的均匀化问题,其中aε(x,λ)是一列快速振荡单调算子且满足文中给出的一致椭圆非一致有界条件.在对这类可能奇异的抛物方程做均匀化时,主要困难来自条件中的‖βε‖L∞(Ω)→+∞,即二阶算子的系数的上界随参数ε→0而趋于+∞.给出最优条件,再仔细结合补偿列紧方法、单调性方法来克服这个困难,得出均匀化结论.

This paper deals with the homogenization of nonlinear parabolic equations of second order：э1uε -divaε（x, Vu,） =fε where aε （x, A ） is a highly oscillating monotonic operator which is uniformly elliptic but non-uniformly bounded. The difficulty of the asymptotic analysis comes from the non-uniform boundedness of βε. An optimal condition on A is given and the classical compensated compactness arguments and monotonic methods are carefully combined to derive the homogenized conclusions.