Martin boundary associated with a system of PDE

Allami Benyaiche; Salma Ghiate

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 3, page 399-425
  • ISSN: 0010-2628

Abstract

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In this paper, we study the Martin boundary associated with a harmonic structure given by a coupled partial differential equations system. We give an integral representation for non negative harmonic functions of this structure. In particular, we obtain such results for biharmonic functions (i.e. $\triangle^{2}\varphi =0$) and for non negative solutions of the equation $\triangle^{2}\varphi =\varphi $.

How to cite

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Benyaiche, Allami, and Ghiate, Salma. "Martin boundary associated with a system of PDE." Commentationes Mathematicae Universitatis Carolinae 47.3 (2006): 399-425. <http://eudml.org/doc/249874>.

@article{Benyaiche2006,
abstract = {In this paper, we study the Martin boundary associated with a harmonic structure given by a coupled partial differential equations system. We give an integral representation for non negative harmonic functions of this structure. In particular, we obtain such results for biharmonic functions (i.e. $\triangle^\{2\}\varphi =0$) and for non negative solutions of the equation $\triangle^\{2\}\varphi =\varphi $.},
author = {Benyaiche, Allami, Ghiate, Salma},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Martin boundary; biharmonic function},
language = {eng},
number = {3},
pages = {399-425},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Martin boundary associated with a system of PDE},
url = {http://eudml.org/doc/249874},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Benyaiche, Allami
AU - Ghiate, Salma
TI - Martin boundary associated with a system of PDE
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 3
SP - 399
EP - 425
AB - In this paper, we study the Martin boundary associated with a harmonic structure given by a coupled partial differential equations system. We give an integral representation for non negative harmonic functions of this structure. In particular, we obtain such results for biharmonic functions (i.e. $\triangle^{2}\varphi =0$) and for non negative solutions of the equation $\triangle^{2}\varphi =\varphi $.
LA - eng
KW - Martin boundary; biharmonic function
UR - http://eudml.org/doc/249874
ER -

References

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  1. Benyaiche A., On almost biharmonic functions in $\Bbb R^n$, Publ. Math., Ec. Norm. Supér, Takaddoum 4 (1988), 47-53 (See Zbl. Math. 678). (1988) 
  2. Benyaiche A., Distributions bi-sousharmoniques sur $\Bbb R^n$ $(n \geq 2)$, Math. Bohem. 119 (1994), 1 1-13. (1994) MR1303547
  3. Benyaiche A., Mesures de représentation sur les espaces biharmoniques, Proc. ICPT 91, Kluwer Acad. Publ., Dordrecht, 1994, pp.171-178. Zbl0824.31006MR1293761
  4. Benyaiche A., Ghiate S., Propriété de moyenne restriente associée à un système d'E.D.P., Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (1) 27 (2003), 125-143. (2003) MR2056415
  5. Benyaiche A., Ghiate S., Frontière de Martin biharmonique, preprint, 2000. 
  6. Benyaiche A., El Gourari A., Caractérisation des ensembles essentiels, J. Math. Soc. Japan 54 2 (2002), 467-486. (2002) MR1883527
  7. Bliedtner J., Hansen W., Potential Theory: An Analytic and Probabilistic Approach to Balayage, Universitext, Springer, Berlin, 1986. Zbl0706.31001MR0850715
  8. Boukricha A., Espaces biharmoniques, Théorie du potentiel (Orsay, 1983), Lecture Notes in Math. 1096, Springer, Berlin, 1984, pp.116-149. Zbl0567.31006MR0890356
  9. Bouleau N., Espaces biharmoniques et couplage de processus de Markov, J. Math. Pures Appl. 59 (1980), 187-240. (1980) Zbl0403.60068MR0581988
  10. Brelot M., On Topologies and Boundaries in Potential Theory, Lecture Notes in Math. 175, Springer, Berlin, 1971. Zbl0277.31002MR0281940
  11. Brelot M., Eléments de la théorie classique du potentiel, 4ème edition, Centre de Documentation Universitaire, Paris, 1969. Zbl0116.07503MR0106366
  12. Constantinescu C., Cornea A., Potential Theory on Harmonic Spaces, Springer, New York-Heidelberg, 1972. Zbl0248.31011MR0419799
  13. Doob J.L., Classical Potential Theory and its Probabilistic Counterpart, Springer, New York, 1984. Zbl0990.31001MR0731258
  14. Hansen W., Modification of balayage spaces by transitions with application to coupling of PDE's, Nagoya Math. J. 169 (2003), 77-118. (2003) Zbl1094.31005MR1962524
  15. Helms L.L., Introduction to Potential Theory, Wiley, New York, 1969. Zbl0188.17203MR0261018
  16. Phelps R.R., Lectures on Choquet's Theorem, Van Nostrand, Princeton-Toronto-London, 1966. Zbl0997.46005MR0193470
  17. Hervé R.M., Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier 12 (1962), 415-571. (1962) MR0139756
  18. Hervé R.M., Hervé M., Les fonctions surharmoniques associées à un opérateur elliptique du second ordre à coefficients discontinus, Ann. Inst. Fourier 19 1 (1969), 305-359. (1969) MR0261027
  19. Meyer M., Balayage spaces on topological sums, Potential Theory (Prague, 1987), Plenum, New York, 1988, pp. 237-246. Zbl0691.31004MR0986301
  20. Nakai M., Martin boundary over an isolated singularity of rotation free density, J. Math. Soc. Japan 26 (1974), 483-507. (1974) Zbl0281.30012MR0361119
  21. Smyrnelis E.P., Axiomatique des fonctions biharmoniques, I, Ann. Inst. Fourier 26 1 35-97 (1976). (1976) MR0477101
  22. Smyrnelis E.P., Axiomatique des fonctions biharmoniques, II, Ann. Inst. Fourier 26 3 1-47 (1976). (1976) MR0477101

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