Hostname: page-component-7b7567c8c-2rsb6 Total loading time: 0 Render date: 2026-11-13T11:21:57.619Z Has data issue: false hasContentIssue false
  • English
  • Français

The Dirichlet Problem for the Slab with Entire Data and a Difference Equation for Harmonic Functions

Published online by Cambridge University Press:  20 November 2018

Dmitry Khavinson,
Erik Lundberg and
Hermann Render
Dmitry Khavinson
Affiliation:
Dept. of Mathematics and Statistics, University of South Florida, Tampa, FL, USA e-mail: dkhavins@usf.edu
Erik Lundberg
Affiliation:
Dept. of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL, USA e-mail: elundber@fau.edu
Hermann Render
Affiliation:
School of Mathematics and Statistics, University College Dublin, Dublin, Ireland e-mail: hermann.render@ucd.ie
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.

It is shown that the Dirichlet problem for the slab $\left( a,\,b \right)\,\times \,{{\mathbb{R}}^{d}}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function $g$ , the inhomogeneous difference equation $h\left( t\,+\,1,\,y \right)\,-\,h\left( t,\,y \right)\,=\,g\left( t,\,y \right)$ has an entire harmonic solution $h$ .

Keywords

31B2031B05reflection principleentire harmonic functionanalytic continuation

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017, pp. 146 - 153
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Appell, P., Sur les fonctions périodiques de deux variables. J. Math. Pures et Appl. 7(1891), 157–219.Google Scholar
[2] Armitage, D. H., The Dirichlet problem when the boundary function is entire. J. Math. Anal. Appl. 291(2004), no. 2, 565–577. http://dx.doi.Org/1 0.101 6/j.jmaa.2003.11.01 6 Google Scholar
[3] Armitage, D. H. and Gardiner, S. J., Classical potential theory. Springer Monographs in Mathematics, Springer-Verlag, London, 2001. http://dx.doi.Org/10.1007/978-1-4471 -0233-5 Google Scholar
[4] Berenstein, C. A. and Gay, R., Complex analysis and special topics in harmonic analysis. Springer-Verlag, New York, 1995. http://dx.doi.Org/10.1007/978-1-4613-8445-8 Google Scholar
[5] Brawn, F. T., The Poisson integral and harmonic majorization in R” x ]0,1[. J. London Math. Soc. 3(1971), 747–760. http://dx.doi.Org/10.1112/jlms/s2-3.4.747 Google Scholar
[6] Durand, W., On some boundary value problems on a strip in the complex plane. Rep. Math. Phys. 52(2003), no. 1, 1-23. http://dx.doi.Org/10.1016/S0034-4877(03)90001-2 Google Scholar
[7] Ebenfelt, P., Khavinson, D., and Shapiro, H. S., Algebraic aspects of the Dirichlet problem. In: Quadrature domains and their applications, Oper. Theory Adv. Appl., 156, Birkhâuser, Basel, 2005, pp. 151-172. http://dx.doi.Org/10.1007/3-7643-7316-4_7 Google Scholar
[8] Friedman, A., On n-metaharmonic functions and functions of infinite order. Proc. Amer. Math. Soc. 8(1957), 223–229.Google Scholar
[9] Gardiner, S. J., The Dirichlet and Neumann problems for harmonic functions in half-spaces. J. London Math. Soc. 24(1981), no. 3, 502-512. http://dx.doi.Org/10.1112/jlms/s2-24.3.5O2 Google Scholar
[10] Gardiner, S. J., The Dirichlet problem with non-compact boundary. Math. Z. 213(1993), no. 1,163-170. http://dx.doi.Org/10.1007/BF03025715 Google Scholar
[11] Gardiner, S. J. and Render, H., Harmonic functions which vanish on a cylindrical surface. J. Math. Anal. Appl. 433(2016), no. 2, 1870–1882. http://dx.doi.Org/10.101 6/j.jmaa.2O1 5.08.077 Google Scholar
[12] Hôrmander, L., The analysis of linear partial differential operators. Vol. I, Springer-Verlag, Berlin-Heidelberg, New York, 1983.Google Scholar
[13] Hurwitz, A., Sur l'intégrale finie d'une fonction entière. Acta Math. 20(1897), no. 1, 285–312. http://dx.doi.Org/1 0.1007/BF0241 8035 Google Scholar
[14] Khavinson, D., Singularities of harmonie functions in C”. In: Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., 52, Amer. Math. Soc, Providence, RI, 1991, pp. 207–217. http://dx.doi.Org/10.1O9O/pspum/O52.3/1128595 Google Scholar
[15] Khavinson, D., Holomorphic partial differential equations and classical potential theory. Universidad de La Laguna, Departamento de Anâlisis Matemâtico, La Laguna 1996.Google Scholar
[16] Khavinson, D., Cauchy's problem for harmonie functions with entire data on a sphere. Canad. Math. Bull. 40(1997), no. 1, 60–66. http://dx.doi.Org/10.4153/CMB-1997-007-3 Google Scholar
[17] Khavinson, D. and Lundberg, E., The search for singularities of solutions to the Dirichlet problem: recent developments. In: Hilbert spaces of analytic functions, CRM Proc. Lecture Notes, 51, American Mathematical Society, Providence, RI, 2010, pp. 121–132.Google Scholar
[18] Khavinson, D. and Shapiro, H. S., Dirichlet's problem when the data is an entire function. Bull. London Math. Soc. 24(1992), no. 5, 456–468. http://dx.doi.Org/10.1112/blms/24.5.456 Google Scholar
[19] Khavinson, D., Lundberg, E., and Render, H., Dirichlet's problem with entire data posed on an ellipsoidal cylinder. arxiv:1 602.0283.Google Scholar
[20] Miyamoto, I., A type of uniqueness of solutions for the Dirichlet problem on a cylinder. Tôhoku Math. J. 48(1996), no. 2, 267–292. http://dx.doi.Org/10.2748/tmj711 78225381 Google Scholar
[21] Render, H., Real Bargmann spaces, Fischer decompositions and sets of uniqueness for polyharmonic functions. Duke Math. J. 142(2008), no. 2, 313–351. http://dx.doi.Org/10.1215/00127094-2008-008 Google Scholar
[22] Siegel, D. and Talvila, E. O., Uniqueness for the n-dimensional half space Dirichlet problem. Pacific J. Math. 175(1996), no. 2, 571–587. http://dx.doi.Org/10.21 40/pjm.1 996.1 75.571 Google Scholar
[23] Shapiro, H. S., An algebraic theorem ofE. Fischer and the holomorphic Goursat problem. Bull. London Math. Soc. 21(1989), no. 6, 513–537. http://dx.doi.Org/10.1112/blms/21.6.513 Google Scholar
[24] Widder, D. V., Functions harmonie in a strip. Proc. Amer. Math. Soc. 12(1961), 67–72. http://dx.doi.Org/10.1090/S0002-9939-1961-0132838-8 Google Scholar