×

zbMATH — the first resource for mathematics

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)

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.