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Positivity for a strongly coupled elliptic system by Green function estimates. (English) Zbl 0792.35048
Uniform positivity for elliptic systems on bounded domains that are coupled by first order derivatives (for example \(-\Delta u = f - q \cdot \nabla v\), \(-\Delta v = au\)) cannot be obtained by the classical maximum principle. However, using the pointwise estimates \[ | \int_ \Omega G(x,z)a(z)G(z,y)dz| \leq c_ 1 G(x,z)\text{ and }| \int_ \Omega G(x,z) q(z) \cdot \nabla_ z G(z,y)dz| \leq c_ 2 G(x,z), \] where \(G(\cdot,\cdot)\) denotes the Green function, one may show for the above example that if \(a\) and \(q\) are small enough, then \(f > 0\) implies \(u > 0\). Such estimates are given for \(a\) and \(q\) in appropriate Schechter type spaces. More general systems are considered, both in dimensions \(n = 2\) and \(n \geq 3\). The estimates are also used to obtain pointwise bounds for the lifetime of a conditioned brownian motion.
Reviewer: G.Sweers (Delft)

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B50 Maximum principles in context of PDEs
45M20 Positive solutions of integral equations
35B45 A priori estimates in context of PDEs
Full Text: DOI
[1] Ancona, A. Comparaison des mesures harmoniques et des fonctions de Green pour des opérateurs elliptiques sur un domaine lipschitzien.C. R. Acad. Sci. Paris 294, 505–508 (1982). · Zbl 0504.35037
[2] Bonnet, A. Une propriété liée à la positivité pour l’opérateur biharmonique. Preprint (1989).
[3] Caristi, G., and Mitidieri, E. Maximum principles for a class of non cooperative elliptic systems.Delft Progress Report 14, 33–56 (1990). · Zbl 0703.35025
[4] Caristi, G., and Mitidieri, E. Further results on maximum principles for non cooperative elliptic systems.Nonl. Ana. T.M.A. 17, 547–558(1991). · Zbl 0766.35003
[5] Courant, R.Dirichlet’s Principle, Conformai Mapping and Minimal Surfaces. New York: Interscience 1950. · Zbl 0040.34603
[6] Cranston, M., and McConnell, T. R. The lifetime of conditioned Brownian motion.Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 1–11 (1983). · Zbl 0506.60071
[7] Cranston, M. Lifetime of conditioned Brownian motion in Lipschitz domains.Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 335–340 (1985). · Zbl 0581.60062
[8] Cranston, M., Fabes, E., and Zhao, Z. Conditional Gauge and potential theory for the Schrödinger operator.Trans. Amer. Math. Soc. 307, 171–194 (1988). · Zbl 0652.60076
[9] Cranston, M., and Zhao, Z. Conditional transformation of drift Formula and potential theory for 1/2 {\(\Delta\)}+b(.).Commun. Math. Phys. 112, 613–625 (1987). · Zbl 0647.60071
[10] De Figueiredo, D., and Mitidieri, E. Maximum principles for linear elliptic systems.Rend. 1st. Mat. Univ. Trieste 24 (1992). · Zbl 0793.35011
[11] Doob, J. L. Conditional Brownian motion and the boundary limits of harmonic functions.Bull. Soc. Math. France 85, 431–458(1957). · Zbl 0097.34004
[12] Gilbarg, D., and Trudinger, N. S.Elliptic partial differential equations of second order. New York: Springer-Verlag 1977. · Zbl 0361.35003
[13] Hille, E.Analytic Function Theory, Vol. II. Waltham, MA: Blaisdell 1962. · Zbl 0102.29401
[14] Hueber, H., and Sieveking, M. Uniform bounds for quotients of Green functions on C1,1 -domains.Ann. Inst. Fourier 32, 105–117(1982). · Zbl 0465.35028
[15] Hueber, H. A uniform estimate for Green functions onC 1,1-domains. Preprint, Universität Bielefeld / Forschungszentrum Bielefeld-Bochum-Stochastik.
[16] Protter, M., and Weinberger, H.Maximum Principles in Differential Equations. Prentice Hall 1976. · Zbl 0153.13602
[17] Simon, B. Schrödinger semigroups.Bull. Amer. Math. Soc. 7, 447–526 (1982). · Zbl 0524.35002
[18] Sweers, G. A strong maximum principle for a noncooperative elliptic system.SIAMJ. Math. Anal. 20, 367–371 (1989). · Zbl 0682.35016
[19] Sweers, G. Strong positivity in \(C(\bar \Omega )\) for elliptic systems.Math. Z. 209, 251–271 (1992). · Zbl 0738.35018
[20] Widman, K.-O. Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations.Math. Scand. 21, 17–37 (1967). · Zbl 0164.13101
[21] Zhao, Z. Green function for Schrödinger operator and conditioned Feynman-Kac gauge.J. Math. Anal. Appl. 116 (1986), 309–334. · Zbl 0608.35012
[22] Zhao, Z. Green functions and conditioned gauge theorem for a 2-dimensional domain. InSeminar on Stochastic Processes, 1987, edited by E. Çinlar et al., pp. 283–294. Boston: Birkhäuser 1988.
[23] Herbst, I. W., and Zhao, Z. Note on a 3G theorem (d = 2). In:Seminar on Stochastic Processes, 1988, edited by E. Çinlar et al., pp. 183–184. Boston: Birkäuser 1989.
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