In this article, we consider the set F of functions annihilated by a uniformly elliptic system S in an open set Ω 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:
|∇k 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 Ω.
As a consequence, we can compare ||u||Lp(Ω) and ||∇ku||Lp(Ω,δkp) where δ is the distance to the boundary of Ω, under the hypothesis that S has constant coefficients or satisfies S(1) = 0.
We conclude that, with the metric ||u||Lp(Ω) + ||∇u||Lp(Ω) we have a compacity property of the ball of F for all p > 0.
@article{urn:eudml:doc:41510,
title = {Propri\'et\'es de moyenne pour les solutions de syst\`emes elliptiques},
journal = {Publicacions Matem\`atiques},
volume = {37},
year = {1993},
pages = {83-89},
mrnumber = {MR1240924},
zbl = {0792.35046},
language = {fr},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41510}
}
Détraz, Jacqueline. Propriétés de moyenne pour les solutions de systèmes elliptiques. Publicacions Matemàtiques, Tome 37 (1993) pp. 83-89. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41510/