Resumen de Propriétés de Moyenne pour les solutions de systèmes elliptiques

Jacqueline Détraz

• In this article, we consider the set F of functions annihilated by a uniformly elliptic system S in an open set O of Rn.

  We show that, as in the case of harmonic functions, F satisfies a submean-property, first for p=2 by elliptic estimates, then for all p > 0:

  |Ñk u(x)|p = C / (rn+kp) ?B(x,r) |u(y)|p dy for each u in F, each k > 0 and every ball B(x,r) included in O.

  As a consequence, we can compare ||u||Lp(O) and ||Ñku||Lp(O,dkp) where d is the distance to the boundary of O, under the hypothesis that S has constant coefficients or satisfies S(1) = 0.

  We conclude that, with the metric ||u||Lp(O) + ||Ñu||Lp(O) we have a compacity property of the ball of F for all p > 0.