Div-Curl lemma and precompactness in $H^{-1}$

Question

I am trying to understand $\operatorname{div-curl}$ lemma. An important requirement to apply $\operatorname{div-curl}$ lemma is the precomapctness of the sequences $\operatorname{div}(A_n)$ and $\operatorname{curl}(B_n)$ in $H^{-1}(\Omega).$

What are the important compactness results which help in checking whether a sequence is precompact in $H^{-1}(\Omega)$ or not? Where can I find the details?

Are there any books which illustrate the application of $\operatorname{div-curl}$ through some examples?

Any help is appreciated..

Answer 1

Usually, div curl lemma is applied for functions $A_n$ and $B_n$ that verifies $div(A_n)$ and $curl(B_n)$ are bounded sequences in $L^2(\Omega)$, which implies the precompactness in $H^{-1}(\Omega)$.

There are several applications to the Div Curl lemma, the most famous is the application to Homogeneization of PDE using Tartar's oscillating test function method. You can find informations easily on the web on this methid. Even if not using Tartar's method, div curl lemma is very often used in Homogeneization.

Hope this help a little.