Abstract
The Schwarz reflection principle applies to a harmonic function which continuously vanishes on a relatively open subset of a planar or spherical boundary surface. It yields a harmonic extension to a predefined larger domain and provides a simple formula for this extension. Although such a point-to-point reflection law is unavailable for other types of surface in higher dimensions, it is natural to investigate whether similar harmonic extension results still hold. This article describes recent progress on such results for the particular case of cylindrical surfaces, and concludes with several open questions.
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References
Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer, London (2001)
Carslaw, H.S.: Integral equations and the determination of Green’s functions in the theory of potential. Proc. Edinb. Math. Soc. 31, 71–89 (1913)
Ebenfelt, P., Khavinson, D.: On point to point reflection of harmonic functions across real-analytic hypersurfaces in \( \mathbb{R}^{n}\). J. Anal. Math. 68, 145–182 (1996)
Gardiner, S.J., Render, H.: Harmonic functions which vanish on a cylindrical surface. J. Math. Anal. Appl. 433, 1870–1882 (2016)
Gardiner, S.J., Render, H.: A reflection result for harmonic functions which vanish on a cylindrical surface. J. Math. Anal. Appl. 443, 81–91 (2016)
Gardiner, S.J., Render, H.: Harmonic functions which vanish on coaxial cylinders. J. Anal. Math. (to appear). ArXiv:1705.09237
Khavinson, D.: Holomorphic Partial Differential Equations and Classical Potential Theory. Universidad de La Laguna, Departamento de Análisis Matemático, La Laguna (1996)
Khavinson, D., Lundberg, E., Render, H.: Dirichlet’s problem with entire data posed on an ellipsoidal cylinder. Potential Anal. 46, 55–62 (2017)
Khavinson, D., Shapiro, H.S.: Remarks on the reflection principle for harmonic functions. J. Anal. Math. 54, 60–76 (1990)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)
Wimp, J., Colton, D.: Remarks on the representation of zero by solutions of differential equations. Proc. Am. Math. Soc. 74, 232–234 (1979)
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Gardiner, S.J., Render, H. Extension results for harmonic functions which vanish on cylindrical surfaces. Anal.Math.Phys. 8, 213–220 (2018). https://doi.org/10.1007/s13324-018-0213-0
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DOI: https://doi.org/10.1007/s13324-018-0213-0