BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, p. 1489-1505
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.
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.
DOI : https://doi.org/10.1016/j.anihpc.2010.09.005
Classification:  49J20,  30L99,  31C45,  31E05,  35J20,  49J27
Keywords: 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},
     publisher = {Elsevier},
     volume = {27},
     number = {6},
     year = {2010},
     pages = {1489-1505},
     doi = {10.1016/j.anihpc.2010.09.005},
     zbl = {1219.49003},
     mrnumber = {2738330},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 27 (2010) no. 6, pp. 1489-1505. doi : 10.1016/j.anihpc.2010.09.005. http://www.numdam.org/item/AIHPC_2010__27_6_1489_0/

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