weak L¹-convergence

Let , a sequence converges weakly to if

As usual the convergence is denoted by $u_n \rightharpoonup u$ in $L^1(\Omega)$.

Definition (equiintegrability). For a family of integrable functions is equiintegrable if the following two conditions hold

  1. The set is tight, i.e. for any there exists a measurable set with

    This condition is trivially true if .
  2. For any there exists such that for every measurable set with

Lemma (Equivalent characterisation of equiintegrability). Let , then is a family of equiintegrable functions if and only if

  1. the family is tight and
  2. there exists an increasing superlinear function such that

Theorem (Dunford-Pettis). A sequence converges weakly in if and only if

  1. the sequence is is equibounded in :

  2. and the sequence is equiintegrable.

Lemma (weak lower semicontinuity of convex functions). If is convex and

then

References

  1. K. H. Karlsen, “Notes on weak convergence (MAT4380 - Spring 2006).” pp. 1–14, 2006
  2. L. C. Evans."Weak convergence methods for nonlinear partial differential equations", volume 74 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990.

Leave a Reply

Your email address will not be published. Required fields are marked *