no. 6
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.

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.

DOI : 10.1016/j.anihpc.2010.09.005
Classification : 49J20, 30L99, 31C45, 31E05, 35J20, 49J27
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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1016/j.anihpc.2010.09.005/

[1] A. Björn, A weak Kellogg property for quasiminimizers, Comment. Math. Helv. 81 (2006), 809-825 | MR | Zbl

[2] A. Björn, Removable singularities for bounded p-harmonic and quasi(super)harmonic functions on metric spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), 71-95 | EuDML | MR | Zbl

[3] A. Björn, A regularity classification of boundary points for p-harmonic functions and quasiminimizers, J. Math. Anal. Appl. 338 (2008), 39-47 | MR | Zbl

[4] A. Björn, Cluster sets for Sobolev functions and quasiminimizers, J. Anal. Math., in press. | MR

[5] A. Björn, J. Björn, Boundary regularity for p-harmonic functions and solutions of the obstacle problem, preprint, LiTH-MAT-R-2004-09, Linköping, 2004.

[6] A. Björn, J. Björn, Boundary regularity for p-harmonic functions and solutions of the obstacle problem, J. Math. Soc. Japan 58 (2006), 1211-1232 | MR | Zbl

[7] A. Björn, J. Björn, Power-type quasiminimizers, Ann. Acad. Sci. Fenn. Math., in press. | MR

[8] A. Björn, J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts Math., European Math. Soc., Zurich, in press. | MR

[9] A. Björn, N. Marola, Moser iteration for (quasi)minimizers on metric spaces, Manuscripta Math. 121 (2006), 339-366 | MR | Zbl

[10] A. Björn, O. Martio, Pasting lemmas and characterizations of boundary regularity for quasiminimizers, Results Math. 55 (2009), 265-279 | MR | Zbl

[11] J. Björn, Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math. 46 (2002), 383-403 | MR | Zbl

[12] J. Björn, Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations, Calc. Var. Partial Differential Equations 35 (2009), 481-496 | MR | Zbl

[13] J. Björn, N. Shanmugalingam, Poincaré inequalities, uniform domains and extension properties for Newton–Sobolev functions in metric spaces, J. Math. Anal. Appl. 332 (2007), 190-208 | MR | Zbl

[14] J. Cheeger, Differentiability of Lipschitz functions on metric spaces, Geom. Funct. Anal. 9 (1999), 428-517 | MR | Zbl

[15] E. Dibenedetto, N.S. Trudinger, Harnack inequalities for quasiminima of variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 295-308 | EuDML | Numdam | MR | Zbl

[16] J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics vol. 29, Amer. Math. Soc., Providence, RI (2001) | MR

[17] B. Franchi, C.E. Gutiérrez, R.L. Wheeden, Weighted Sobolev–Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), 523-604 | MR | Zbl

[18] J. García-Cuerva, J.L. Rubio De Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam (1985) | MR | Zbl

[19] M. Giaquinta, E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31-46 | MR | Zbl

[20] M. Giaquinta, E. Giusti, Quasi-minima, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 79-107 | EuDML | Numdam | MR | Zbl

[21] P. Hajłasz, P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris Ser. I Math. 320 (1995), 1211-1215 | MR | Zbl

[22] P. Hajłasz, P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 no. 688 (2000) | MR | Zbl

[23] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York (2001) | MR | Zbl

[24] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover, Mineola, NY (2006) | MR | Zbl

[25] J. Heinonen, P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61 | MR | Zbl

[26] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523 | MR | Zbl

[27] P.T. Judin, Onedimensional Quasiminimizers and Quasisuperminimizers [Yksiulotteiset Kvasiminimoijat ja Kvasisuperminimoijat], Licentiate thesis, Dept. of Math., Helsinki University, Helsinki, 2006 (in Finnish).

[28] J. Kinnunen, O. Martio, Potential theory of quasiminimizers, Ann. Acad. Sci. Fenn. Math. 28 (2003), 459-490 | EuDML | MR | Zbl

[29] J. Kinnunen, O. Martio, Sobolev space properties of superharmonic functions on metric spaces, Results Math. 44 (2003), 114-129 | MR | Zbl

[30] J. Kinnunen, N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001), 401-423 | MR | Zbl

[31] P. Koskela, P. Macmanus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1-17 | EuDML | MR | Zbl

[32] O.E. Maasalo, Global integrability of p-superharmonic functions on metric spaces, J. Anal. Math. 106 (2008), 191-207 | MR | Zbl

[33] N. Marola, Moser's method for minimizers on metric measure spaces, preprint A478, Helsinki University of Technology, Institute of Mathematics, Helsinki, 2004.

[34] O. Martio, Boundary behavior of quasiminimizers and Dirichlet finite PWB solutions in the borderline case, Report in Math. 440, University of Helsinki, Helsinki, 2006.

[35] O. Martio, Reflection principle for quasiminimizers, Funct. Approx. Comment. Math. 40 (2009), 165-173 | MR | Zbl

[36] O. Martio, Capacity and potential estimates for quasiminimizers, Complex Anal. Oper. Theory, doi:10.1007/s11785-010-0074-5, in press. | MR

[37] O. Martio, Quasiminimizers – definitions, constructions and capacity estimates, http://www.mai.liu.se/TM/conf09/martio.pdf

[38] O. Martio, C. Sbordone, Quasiminimizers in one dimension: integrability of the derivative, inverse function and obstacle problems, Ann. Mat. Pura Appl. 186 (2007), 579-590 | MR | Zbl

[39] H.M. Reimann, T. Rychener, Funktionen beschränkter mittlerer Oszillation, Lecture Notes in Mathematics vol. 487, Springer, Berlin (1975) | MR | Zbl

[40] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243-279 | EuDML | MR | Zbl

[41] N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), 1021-1050 | MR | Zbl

[42] P. Tolksdorf, Remarks on quasi(sub)minima, Nonlinear Anal. 10 (1986), 115-120 | MR | Zbl

[43] N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747 | MR | Zbl

[44] H. Uppman, The reflection principle for one-dimensional quasiminimizers, http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-19162

[45] W.P. Ziemer, Boundary regularity for quasiminima, Arch. Rational Mech. Anal. 92 (1986), 371-382 | MR | Zbl

Cité par Sources :