G-CONVERGENCE FOR NON-DIVERGENCE SECOND-ORDER ELLIPTIC OPERATORS IN THE PLANE

The central theme of this paper is the study of G-convergence of elliptic operators in the plane. We consider the operator M[u] = Tr (A(z)D(2)u) = a(11)(z)u(xx) + 2a(12)(z)u(xy) + a(22)(z)u(yy) and its formal adjoint N[v] = D-2(A(w)v) = (a(11)(w)v)(xx) + 2(a(12)(w)v)(xy) + (a(22)(w)v)(yy), where u is an element of W-2,W-P and v is an element of L-P, with p > 1, and A is a symmetric uniformly bounded elliptic matrix such that det A = 1 almost everywhere. We generalize a theorem due to Sirazhudinov Zhikov, which is a counterpart of the Div-Curl lemma for elliptic operators in non-divergence form. As an application, under suitable assumptions, we characterize the G-limit of a sequence of elliptic operators. In the last section we consider elliptic matrices whose coefficients are also in VMO; this leads us to extend our result to any exponent p is an element of (1, 2).