Compensated Compactness and Oscillating Test Functions

Abstract

While Chapter 4 provides a very efficient method to construct formal asymptotic expansions of solutions in multiscale problems, direct justification of such formal expansions can become rather cumbersome. This chapter is devoted to an alternative justification approach that is much simpler to implement. It is based on the use of specially chosen oscillating test functions in the weak formulation of the problem so that upon passing to the limit one obtains the weak formulation of the homogenized problem. This passage to the limit is nontrivial as products of weakly converging sequences are typically involved. We formulate and prove the celebrated Div-Curl Lemma which allows one to pass to the limit in products of weakly converging sequences.