Analysis of Divergence-Free Parametrized Measures

Abstract

The question we address in this chapter is the characterization of parametrized measures coming from sequences of vector-valued functions u _j Ω ⊂ R _N → R _m uniformly bounded in L ^∞(Ω) for which we have additional information in the form

$$ \left\{ {Au_j } \right\} relatively compact in H^{ - 1} \left( \Omega\right) $$

(10-1)

, for A a differential operator of type

$${{\left( {Au} \right)}_{i}} = \sum\limits_{{l,k}} {ailk\frac{{\partial {{u}^{l}}}}{{\partial {{x}_{k}}}},\;\;\;i = 1, \ldots ,s,}$$

with constant coefficientsa _ilkIn the previous chapters, we have concentrated on the fundamental case when A = curl, m replaced by m × N,

$$curl\;u = \frac{{\partial u_{l}^{i}}}{{\partial {{x}_{k}}}} - \frac{{\partial u_{k}^{i}}}{{\partial {{x}_{1}}}},\;\;i = 1, \ldots ,m.\;\;l,k = 1, \ldots ,N,$$

and we asked forAu =0 rather than (10-1). In general, let

$$ \vartheta= \left\{ {(\lambda ,\xi ) \in R^m\times R^N :\sum\limits_{l,k} {a_{ilk} \lambda _l } \xi _k= 0} \right\} $$

and let the characteristic cone be defined by

$$ \Lambda= \left\{ {\lambda\in R^m :\:there\:is\:a\:\xi\in R^N- \{ 0\} ,(\lambda ,\xi ) \in \vartheta } \right\} $$

.