Harmonic extension from the exterior of a cylinder

Gardiner, Stephen; Render, Hermann

doi:10.1090/proc/15172

Harmonic extension from the exterior of a cylinder
HTML articles powered by AMS MathViewer

by Stephen J. Gardiner and Hermann Render PDF

Proc. Amer. Math. Soc. 149 (2021), 1077-1089 Request permission

Abstract:

Let $h$ be a harmonic function on an annular cylinder. It was recently shown that, if $h$ vanishes on the outer cylindrical boundary component, then it has a harmonic extension to a larger annular cylinder formed by radial reflection. This paper shows that a different kind of extension result holds if $h$ instead vanishes on the inner boundary component. As a corollary it is shown that any harmonic function on the exterior of an infinite cylinder which vanishes on the boundary has a harmonic extension to the complement of the axis of the cylinder.

References

David H. Armitage and Stephen J. Gardiner, Classical potential theory, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001. MR 1801253, DOI 10.1007/978-1-4471-0233-5
Árpád Baricz, Dragana Jankov Maširević, Saminathan Ponnusamy, and Sanjeev Singh, Bounds for the product of modified Bessel functions, Aequationes Math. 90 (2016), no. 4, 859–870. MR 3523103, DOI 10.1007/s00010-016-0414-2
J. Dougall, The determination of Green’s function by means of cylindrical or spherical harmonics, Proc. Edinb. Math. Soc. 18 (1900), 33-83.
Peter Ebenfelt and Dmitry Khavinson, On point to point reflection of harmonic functions across real-analytic hypersurfaces in $\mathbf R^n$, J. Anal. Math. 68 (1996), 145–182. MR 1403255, DOI 10.1007/BF02790208
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, Vol. II. McGraw-Hill, New York, 1953.
Stephen J. Gardiner and Hermann Render, Harmonic functions which vanish on a cylindrical surface, J. Math. Anal. Appl. 433 (2016), no. 2, 1870–1882. MR 3398795, DOI 10.1016/j.jmaa.2015.08.077
Stephen J. Gardiner and Hermann Render, A reflection result for harmonic functions which vanish on a cylindrical surface, J. Math. Anal. Appl. 443 (2016), no. 1, 81–91. MR 3508480, DOI 10.1016/j.jmaa.2016.05.007
Stephen J. Gardiner and Hermann Render, Extension results for harmonic functions which vanish on cylindrical surfaces, Anal. Math. Phys. 8 (2018), no. 2, 213–220. MR 3817851, DOI 10.1007/s13324-018-0213-0
Stephen J. Gardiner and Hermann Render, Harmonic functions which vanish on coaxial cylinders, J. Anal. Math. 138 (2019), no. 2, 891–915. MR 3996062, DOI 10.1007/s11854-019-0050-6
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
Dmitry Khavinson and Erik Lundberg, Linear holomorphic partial differential equations and classical potential theory, Mathematical Surveys and Monographs, vol. 232, American Mathematical Society, Providence, RI, 2018. MR 3821527, DOI 10.1090/surv/232
Dmitry Khavinson and Harold S. Shapiro, Remarks on the reflection principle for harmonic functions, J. Analyse Math. 54 (1990), 60–76. MR 1041175, DOI 10.1007/BF02796142
Maciej Klimek, Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1150978
N. N. Lebedev, Special functions and their applications, Revised English edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. Translated and edited by Richard A. Silverman. MR 0174795
Diego Ruiz-Antolín and Javier Segura, A new type of sharp bounds for ratios of modified Bessel functions, J. Math. Anal. Appl. 443 (2016), no. 2, 1232–1246. MR 3514344, DOI 10.1016/j.jmaa.2016.06.011
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746