Researches On Two Types Of Equations With Curl Operators

This thesis devotes to the homogenization of two types of differential equations with curl operators. The first type of equations where A(x, y) is a positive definite matrix, which is Y-periodic with respect to y-variable, Y is the unit cube inR3, A(x, y)∈C(Ω; Cp(Y))3×3,f∈H(Ω, div). We apply the method of Two-scale Convergence to discuss the homogeniza-tion of two cases:i) r=1; ii) r>1. And then we give a description of the L2weak limits of solutions uε.The second type is the equations Matrix Aε∈L2(Q)3×3, satisfies: μ1|&|2≤ζ·Aεζ≤μ2|ζ|2, where0<μ1≤μ2,(？)ζ∈R3.f∈Ho(Ω,curl)’, divf=0. For this type of equations with curl operators, we extend the definition of G-convergence introduced by S. Spagnolo. Several propositions of extended G-convergence are established. Furthermore, we give a sufficient condition of extended G-convergence.