| Title:
|
On the potential theory of some systems of coupled PDEs (English) |
| Author:
|
Aslimani, Abderrahim |
| Author:
|
El Ghazi, Imad |
| Author:
|
El Kadiri, Mohamed |
| Author:
|
Haddad, Sabah |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
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1213-7243 (online) |
| Volume:
|
57 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
135-154 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_1u =-\mu_1v$, $L_2v =-\mu_2u$, on a domain $D$ of $\mathbb R^d$, where $\mu_1$ and $\mu_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class $\mathcal C^\infty$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and a convergence property for increasing sequences of solutions of (S). (English) |
| Keyword:
|
harmonic function |
| Keyword:
|
superharmonic function |
| Keyword:
|
potential |
| Keyword:
|
elliptic linear differential operator |
| Keyword:
|
kernel |
| Keyword:
|
coupled PDEs system |
| Keyword:
|
Kato measure |
| MSC:
|
31B05 |
| MSC:
|
31B10 |
| MSC:
|
31B35 |
| idZBL:
|
Zbl 1363.31004 |
| idMR:
|
MR3513440 |
| DOI:
|
10.14712/1213-7243.2015.165 |
| . |
| Date available:
|
2016-07-05T15:01:29Z |
| Last updated:
|
2018-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145751 |
| . |
| Reference:
|
[1] Armitage D.H., Gardiner S.J.: Classical Potential Theory.Springer, London, 2001. Zbl 0972.31001, MR 1801253 |
| Reference:
|
[2] Bliedtner J., Hansen W.: Potential theory. An analytic and probabilistic approach to balayage.Universitext, Springer, Berlin, 1986. Zbl 0706.31001, MR 0850715 |
| Reference:
|
[3] Boboc N., Bucur Gh.: Perturbations in excessive structures.Complex analysis–fifth Romanian-Finnish seminar, Part 2 (Bucharest, 1981), Lecture Notes in Math., 1014, Springer, Berlin, 1983, pp. 155–187. Zbl 0534.47008, MR 0738120 |
| Reference:
|
[4] Bouleau N.: Semi-groupe triangulaire associé à un espace biharmonique.C.R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 7, A415–A417. Zbl 0405.31009, MR 0552066 |
| Reference:
|
[5] Bouleau N.: Couplage de deux semi-groupes droites.C.R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 8, A465–A467. MR 0527698 |
| Reference:
|
[6] Bouleau N.: Espaces biharmoniques et couplage de processus de Markov.J. Math. Pures Appl. (9) 59 (1980), no. 2, 187–240. Zbl 0403.60068, MR 0581988 |
| Reference:
|
[7] Bouleau N.: Théorie du potentiel associée à certains systèmes différentiels.Math. Ann. 255 (1981), no. 3, 335–350. Zbl 0441.31006, MR 0615854, 10.1007/BF01450707 |
| Reference:
|
[8] Brelot M.: Axiomatique des fonctions harmoniques.Université de Montréal, 1966. Zbl 0148.10401, MR 0247124 |
| Reference:
|
[9] Chen Z.Q., Zhao Z.: Potential theory for elliptic systems.Ann. Probab. 24 (1996), no. 1, 293–319. Zbl 0854.60062, MR 1387637, 10.1214/aop/1042644718 |
| Reference:
|
[10] Constantinescu C.A., Cornea A.: Potential Theory on Harmonic Spaces.Springer, New York-Heidelberg, 1972. Zbl 0248.31011, MR 0419799 |
| Reference:
|
[11] Doob J.L: Classical Potential Theory and its Probabilistic Counterpart.Springer, New York, 1984. Zbl 0990.31001, MR 0731258 |
| Reference:
|
[12] El Kadiri M.: Sur la représentation intégrale en théorie axiomatique des fonctions biharmoniques.Rev. Roumaine Math. Pures Appl. 42 (1997), no. 7–8, 579–589. Zbl 1089.31501, MR 1650389 |
| Reference:
|
[13] El Kadiri M.: Frontière de Martin biharmonique et représentation intégrale des fonctions biharmoniques.Positivity 6 (2002), 129–145. Zbl 0998.31004, MR 1905385, 10.1023/A:1015283920365 |
| Reference:
|
[14] El Kadiri M., Haddad S.: Comportement des fonctions bisurharmoniques et problème de Riquier fin à la frontière de Martin biharmonique.Algebras Groups Geom. 24 (2007), 155–186. Zbl 1148.31007, MR 2345849 |
| Reference:
|
[15] Gazzola F., Sweers G.: On positivity for the biharmonic operator under Steklov boundary conditions.Arch. Ration. Mech. Anal. 188 (2008), no. 3, 399–427. Zbl 1155.35019, MR 2393435, 10.1007/s00205-007-0090-4 |
| Reference:
|
[16] Gazzola F., Grunau H.-C., Sweers G.: Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains.Lecture Notes in Mathematics, 1991, Springer, Berlin, 2010. Zbl 1239.35002, MR 2667016 |
| Reference:
|
[17] Grunau H.-C., Sweers G.: Positivity properties of elliptic boundary value problems of higher order.Proceedings of the Second World Congress of Nonlinear Analysts, Part 8 (Athens, 1996), Nonlinear Anal. 30 (1997), no. 8, 5251–5258. Zbl 0894.35016, MR 1726027 |
| Reference:
|
[18] Grunau H-C., Sweers G.: Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions.Math. Ann. 307 (1997), no. 4, 589–626. Zbl 0892.35031, MR 1464133, 10.1007/s002080050052 |
| Reference:
|
[19] Hansen W.: Harnack inequalities for Schrödinger operators.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 3, 413–470. Zbl 0940.35063, MR 1736524 |
| Reference:
|
[20] Hansen W.: Modification of balayage spaces by transitions with application to coupling of PDE's.Nagoya Math. J. 169 (2003), 77–118. Zbl 1094.31005, MR 1962524, 10.1017/S002776300000845X |
| Reference:
|
[21] Helms L.L.: Introduction to Potential Theory.Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience A Division of John Wiley and Sons, New York-London-Sydney, 1969. Zbl 0188.17203, MR 0261018 |
| Reference:
|
[22] Hervé R.-M.: Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel.Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571. Zbl 0101.08103, MR 0139756, 10.5802/aif.125 |
| Reference:
|
[23] Janssen K.: On the Martin boundary of weakly coupled balayage spaces.Rev. Roumaine Math. Pures Appl. 51 (2006), no. 5–6, 655–664. Zbl 1120.31005, MR 2320915 |
| Reference:
|
[24] Mokobodzki G.: Représentation intégrale des fonctions surharmoniques au moyen des réduites.Ann. Inst. Fourier (Grenoble) 15 (1965), fasc. 1, 103–112. Zbl 0134.09502, MR 0196110, 10.5802/aif.199 |
| Reference:
|
[25] Smyrnélis E.P.: Axiomatique des fonctions biharmoniques. I..Ann. Inst. Fourier (Grenoble) 25 (1975), no. 1, 35–97. Zbl 0295.31006, MR 0382691, 10.5802/aif.544 |
| Reference:
|
[26] Smyrnélis E.P.: Axiomatique des fonctions biharmoniques. II..Ann. Inst. Fourier (Grenoble) 26 (1976), no. 3., 1–47. Zbl 0325.31020, MR 0477101, 10.5802/aif.624 |
| Reference:
|
[27] Sweers G.: Positivity for a strongly coupled elliptic system by Green function estimates.J. Geom. Anal. 4 (1994), no. 1, 121–142. Zbl 0792.35048, MR 1274141, 10.1007/BF02921596 |
| Reference:
|
[28] Sweers G.: Strong positivity in $C(\overline\Omega)$ for elliptic systems.Math. Z. 209 (1992), no. 2, 251–271. MR 1147817, 10.1007/BF02570833 |
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