Dans cet article nous utilisons les propriétés des fonctions avec puissance radiale afin d'obtenir des contre-exemples à certaines inéquations de type Caccioppoli et Harnack faible pour les fonctions quasisuperharmoniques, lesquelles sont bien connues être valables pour les fonctions p-superharmoniques. Nous obtenons aussi de nouvelles bornes pour l'intégrabilité locale des fonctions quasisuperharmoniques. De plus nous démontrons que le logarithme d'une fonction positive quasiminimisante est de type BMO, et appartient à un espace de Sobolev.
In this paper we use quasiminimizing properties of radial power-type functions to deduce counterexamples to certain Caccioppoli-type inequalities and weak Harnack inequalities for quasisuperharmonic functions, both of which are well known to hold for p-superharmonic functions. We also obtain new bounds on the local integrability for quasisuperharmonic functions. Furthermore, we show that the logarithm of a positive quasisuperminimizer has bounded mean oscillation and belongs to a Sobolev type space.
Mots clés : Bounded mean oscillation, Doubling measure, Metric space, Nonlinear, p-harmonic, Poincaré inequality, Potential theory, Quasiminimizer, Quasisuperharmonic, Quasisuperminimizer, Weak Harnack inequality
@article{AIHPC_2010__27_6_1489_0, author = {Bj\"orn, Anders and Bj\"orn, Jana and Marola, Niko}, title = {BMO, integrability, {Harnack} and {Caccioppoli} inequalities for quasiminimizers}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1489--1505}, publisher = {Elsevier}, volume = {27}, number = {6}, year = {2010}, doi = {10.1016/j.anihpc.2010.09.005}, mrnumber = {2738330}, zbl = {1219.49003}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.005/} }
TY - JOUR AU - Björn, Anders AU - Björn, Jana AU - Marola, Niko TI - BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 1489 EP - 1505 VL - 27 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.005/ DO - 10.1016/j.anihpc.2010.09.005 LA - en ID - AIHPC_2010__27_6_1489_0 ER -
%0 Journal Article %A Björn, Anders %A Björn, Jana %A Marola, Niko %T BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 1489-1505 %V 27 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.005/ %R 10.1016/j.anihpc.2010.09.005 %G en %F AIHPC_2010__27_6_1489_0
Björn, Anders; Björn, Jana; Marola, Niko. BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1489-1505. doi : 10.1016/j.anihpc.2010.09.005. http://www.numdam.org/articles/10.1016/j.anihpc.2010.09.005/
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