Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T05:20:50.845Z Has data issue: false hasContentIssue false
  • English
  • Français

The Gradient of a Solution of the Poisson Equation in the Unit Ball and Related Operators

Published online by Cambridge University Press:  20 November 2018

David Kalaj  and
Djordjije Vujadinovic
David Kalaj
Affiliation:
Faculty of Mathematics, University of Montenegro, Dzordza Vašingtona bb, 81000 Podgorica, Montenegro. e-mail: [email protected] e-mail: [email protected]
Djordjije Vujadinovic
Affiliation:
Faculty of Mathematics, University of Montenegro, Dzordza Vašingtona bb, 81000 Podgorica, Montenegro. e-mail: [email protected] e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we determine the ${{L}^{1}}\to {{L}^{1}}$ and ${{L}^{\infty }}\to {{L}^{\infty }}$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

Keywords

35J0547G10Möbius transformationPoisson equationNewtonian potentialCauchy transformBessel function
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 3 , 01 September 2017 , pp. 536 - 545
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Agmon, S., Douglis, A., and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12(1959), 623–727. http://dx.doi.Org/10.1002/cpa.31 60120405 Google Scholar
[2] Ahlfors, L. V., Mb'bius transformations in several dimensions. University of Minnesota, School of Mathematics, 1981. Google Scholar
[3] Anderson, J. M., and A. Hinkkanen, The Cauchy transform on bounded domains. Proc. Amer. Math. Soc. 107(1989), no. 1, 179–185. http://dx.doi.Org/10.1090/S0002-9939-1989-0972226-5 http://dx.doi.Org/10.2307/2048052 Google Scholar
[4] Anderson, J. M., D. Khavinson, and V. Lomonosov, Spectral properties of some integral operators arising in potential theory. Quart. J. Math. Oxford Ser. (2) 43(1992), no. 172, 387–407. http://dx.doi.Org/10.1093/qmathj743.4.387 Google Scholar
[5] Dostanic, M., The properties of the Cauchy transform on a bounded domain, J. Operator Theory 36(1996), 233–247 Google Scholar
[6] Dostanic, M., Norm estimate of the Cauchy transform on Lf(Cl). Integral Equations Operator Theory 52(2005), no. 4, 465–475. http://dx.doi.Org/10.1007/s00020-002-1290-9 Google Scholar
[7] Dostanic, M., Estimate of the second term in the spectral asymptotic of Cauchy transform. J. Funct. Anal. 249(2007), no. 1, 55–74. http://dx.doi.Org/10.101 6/j.jfa.2007.04.007 Google Scholar
[8] Dragomir, S. S., Agarwal, R. P., and Barnett, N. S., Inequalities for Beta and Gamma functions via some classical and new integral inequalities. (English) J. Inequal. Appl. 5(2000), no.2,103-165. http://dx.doi.Org/10.1155/S102 5583400000084 Google Scholar
[9] Gilbarg, D. and Trudinger, N., Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. http://dx.doi.Org/10.1007/978-3-642-61798-0 Google Scholar
[10] Kalaj, D., On some integral operators related to the Poisson equation. Integral Equations Operator Theory 72(2012), 563–575. http://dx.doi.Org/10.1007/s00020-012-1952-1 Google Scholar
[11] Kalaj, D., Cauchy transform and Poisson's equation. Adv. Math. 231(2012), no. 1, 213–242. http://dx.doi.Org/10.1016/j.aim.2012.05.003 Google Scholar
[12] Kalaj, D. and Pavlovic, M., On quasiconformal self-mappings of the unit disk satisfying Poisson's equation. Trans. Amer. Math. Soc. 363(2011), 4043–4061. http://dx.doi.Org/!0.1090/S0002-9947-2011-05081-6 Google Scholar
[13] Kalaj, D. and Vujadinovic, Dj., The solution operator of the inhomogeneous Dirichlet problem in the unit ball. Proc. Amer. Math. Soc. 144(2016), 623–635. http://dx.doi.Org/10.1090/procyi2 723 Google Scholar
[14] Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and series, Elementary Functions, 1. Gordon and Breach, New York, 1986.Google Scholar
[15] Thorin, G., Convexity theorems generalizing those of M. Riesz and Hadamard with some applications. Comm. Sem. Math. Univ. Lund 9(1948), 1–58.Google Scholar