Abstract
It is shown that the Neumann function for −Δu+u=f on a bounded domain Ω⊂ℝn can be estimated pointwise from below in a uniform way. The proof is based on known uniform estimates from below for the Green function.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ancona, A.: Comparaison des mesures harmoniques et des fonctions de Green pour des opérateurs elliptiques sur un domaine lipschitzien. C. R. Acad. Sci., Paris 204, 472–473 (1982)
Chung, K.L., Zhao, Z.: From Brownian motion to Schrödinger’s equation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312. Springer-Verlag, Berlin (1995)
Dall’Acqua, A., Sweers, G.: On domains for which the clamped plate system is positivity preserving. In: Conca, C., Manasevich, R., Uhlmann, G., Vogelius, M. (eds.) Partial Differential Equations and Inverse Problems. Am. Math. Soc., Providence (2004)
Grunau, H.-Ch., Robert, F., Sweers, G.: Optimal estimates from below for biharmonic Green functions. Proc. Am. Math. Soc. 139, 2151–2161 (2010)
Krasovskiĭ, Ju.P.: Isolation of the singularity in Green’s function. (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 31, 977–1010 (1967)
Krasovskiĭ, Ju.P.: Investigation of potentials connected with boundary value problems for elliptic equations. (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 31, 587–640 (1967)
Sweers, G.: Positivity for a strongly coupled elliptic system by Green function estimates. J. Geom. Anal. 4, 121–142 (1994)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, England; The Macmillan Company, New York (1944)
Zhao, Z.: Green function for Schrödinger operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl. 116(2), 309–334 (1986)
Zhao, Z.: Green Functions and Conditioned Gauge Theorem for a 2-Dimensional Domain. Seminar on Stochastic Processes, pp. 283–294. Princeton (1987). Progress in Probability and Statistics, vol. 15. Birkhäuser Boston, Boston (1988)
Acknowledgements
This note would not have been written, if Pavol Quittner had not been asking the right questions. I thank him for the inspiration. I also thank the referee for his comments which thoroughly simplified the proof.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
To the memory of Wolfgang Walter
Rights and permissions
Copyright information
© 2012 Springer Basel
About this paper
Cite this paper
Sweers, G. (2012). Green Function Estimates Lead to Neumann Function Estimates. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0249-9_4
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0248-2
Online ISBN: 978-3-0348-0249-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)