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Keywords:
harmonic function; superharmonic function; potential; elliptic linear differential operator; kernel; coupled PDEs system; Kato measure
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).
References:
[1] Armitage D.H., Gardiner S.J.: Classical Potential Theory. Springer, London, 2001. MR 1801253 | Zbl 0972.31001
[2] Bliedtner J., Hansen W.: Potential theory. An analytic and probabilistic approach to balayage. Universitext, Springer, Berlin, 1986. MR 0850715 | Zbl 0706.31001
[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. MR 0738120 | Zbl 0534.47008
[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. MR 0552066 | Zbl 0405.31009
[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
[6] Bouleau N.: Espaces biharmoniques et couplage de processus de Markov. J. Math. Pures Appl. (9) 59 (1980), no. 2, 187–240. MR 0581988 | Zbl 0403.60068
[7] Bouleau N.: Théorie du potentiel associée à certains systèmes différentiels. Math. Ann. 255 (1981), no. 3, 335–350. DOI 10.1007/BF01450707 | MR 0615854 | Zbl 0441.31006
[8] Brelot M.: Axiomatique des fonctions harmoniques. Université de Montréal, 1966. MR 0247124 | Zbl 0148.10401
[9] Chen Z.Q., Zhao Z.: Potential theory for elliptic systems. Ann. Probab. 24 (1996), no. 1, 293–319. DOI 10.1214/aop/1042644718 | MR 1387637 | Zbl 0854.60062
[10] Constantinescu C.A., Cornea A.: Potential Theory on Harmonic Spaces. Springer, New York-Heidelberg, 1972. MR 0419799 | Zbl 0248.31011
[11] Doob J.L: Classical Potential Theory and its Probabilistic Counterpart. Springer, New York, 1984. MR 0731258 | Zbl 0990.31001
[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. MR 1650389 | Zbl 1089.31501
[13] El Kadiri M.: Frontière de Martin biharmonique et représentation intégrale des fonctions biharmoniques. Positivity 6 (2002), 129–145. DOI 10.1023/A:1015283920365 | MR 1905385 | Zbl 0998.31004
[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. MR 2345849 | Zbl 1148.31007
[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. DOI 10.1007/s00205-007-0090-4 | MR 2393435 | Zbl 1155.35019
[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. MR 2667016 | Zbl 1239.35002
[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. MR 1726027 | Zbl 0894.35016
[18] Grunau H-C., Sweers G.: Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions. Math. Ann. 307 (1997), no. 4, 589–626. DOI 10.1007/s002080050052 | MR 1464133 | Zbl 0892.35031
[19] Hansen W.: Harnack inequalities for Schrödinger operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 3, 413–470. MR 1736524 | Zbl 0940.35063
[20] Hansen W.: Modification of balayage spaces by transitions with application to coupling of PDE's. Nagoya Math. J. 169 (2003), 77–118. DOI 10.1017/S002776300000845X | MR 1962524 | Zbl 1094.31005
[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. MR 0261018 | Zbl 0188.17203
[22] Hervé R.-M.: Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel. Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571. DOI 10.5802/aif.125 | MR 0139756 | Zbl 0101.08103
[23] Janssen K.: On the Martin boundary of weakly coupled balayage spaces. Rev. Roumaine Math. Pures Appl. 51 (2006), no. 5–6, 655–664. MR 2320915 | Zbl 1120.31005
[24] Mokobodzki G.: Représentation intégrale des fonctions surharmoniques au moyen des réduites. Ann. Inst. Fourier (Grenoble) 15 (1965), fasc. 1, 103–112. DOI 10.5802/aif.199 | MR 0196110 | Zbl 0134.09502
[25] Smyrnélis E.P.: Axiomatique des fonctions biharmoniques. I. Ann. Inst. Fourier (Grenoble) 25 (1975), no. 1, 35–97. DOI 10.5802/aif.544 | MR 0382691 | Zbl 0295.31006
[26] Smyrnélis E.P.: Axiomatique des fonctions biharmoniques. II. Ann. Inst. Fourier (Grenoble) 26 (1976), no. 3., 1–47. DOI 10.5802/aif.624 | MR 0477101 | Zbl 0325.31020
[27] Sweers G.: Positivity for a strongly coupled elliptic system by Green function estimates. J. Geom. Anal. 4 (1994), no. 1, 121–142. DOI 10.1007/BF02921596 | MR 1274141 | Zbl 0792.35048
[28] Sweers G.: Strong positivity in $C(\overline\Omega)$ for elliptic systems. Math. Z. 209 (1992), no. 2, 251–271. DOI 10.1007/BF02570833 | MR 1147817
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