Email this article On $C^m$-reflection of harmonic functions and $C^m$-approximation by harmonic polynomials
- Authors: Paramonov P.V.1,2, Fedorovskiy K.Y.2,3
-
Affiliations:
- Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
- Bauman Moscow State Technical University
- Saint Petersburg State University
- Issue: Vol 211, No 8 (2020)
- Pages: 102-113
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133342
- DOI: https://doi.org/10.4213/sm9295
- ID: 133342
Cite item
Open Access
Access granted
Subscription Access
Abstract
We obtain several new sharp $C^m$-continuity conditions, both necessary and sufficient, for operators of harmonic reflection of functions over boundaries of simple Caratheodory domains in $\mathbb R^N$. These results are based on a new criterion (also obtained in this paper) for $C^m$-continuity of the Poisson operator in the aforesaid domains. As corollaries, we give new sufficient conditions for $C^m$-approximability of functions by harmonic polynomials on boundaries of simple Caratheodory domains in $\mathbb R^N$. Bibliography: 17 titles.
About the authors
Petr Vladimirovich Paramonov
Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; Bauman Moscow State Technical University
Email: petr.paramonov@list.ru
Konstantin Yurievich Fedorovskiy
Bauman Moscow State Technical University; Saint Petersburg State University
Email: kfedorovs@yandex.ru
Doctor of physico-mathematical sciences, Associate professor
References
- П. В. Парамонов, “О $operatorname{Lip}^m$- и $C^m$-отражении гармонических функций относительно границ областей Каратеодори в $mathbb R^2$”, Вестн. МГТУ им. Н. Э. Баумана. Сер. Естественные науки, 2018, № 4, 36–45
- K. Fedorovskiy, P. Paramonov, “On $operatorname{Lip}^m$-reflection of harmonic functions over boundaries of simple Caratheodory domains”, Anal. Math. Phys., 9:3 (2019), 1031–1042
- H. Lebesgue, “Sur le problème de Dirichlet”, Rend. Circ. Mat. Palermo, 24 (1907), 371–402
- И. Стейн, Сингулярные интегралы и дифференциальные свойства функций, Мир, М., 1973, 342 с.
- D. H. Armitage, “Reflection principles for harmonic and polyharmonic functions”, J. Math. Anal. Appl., 65:1 (1978), 44–55
- D. Khavinson, H. S. Shapiro, “Remarks on the reflection principle for harmonic functions”, J. Anal. Math., 54 (1990), 60–76
- D. Khavinson, “On reflection of harmonic Functions in surfaces of revolution”, Complex Variables Theory Appl., 17:1-2 (1991), 7–14
- P. Ebenfelt, D. Khavinson, “On point-to-point reflection of harmonic functions across real-analytic hypersurfaces in $mathbb R^n$”, J. Anal. Math., 68 (1996), 145–182
- S. J. Gardiner, H. Render, “A reflection result for harmonic functions which vanish on a cylindrical surface”, J. Math. Anal. Appl., 443:1 (2016), 81–91
- E. Schippers, W. Staubach, “Harmonic reflection in quasicircles and well-posedness of a Riemann–Hilbert problem on quasidisks”, J. Math. Anal. Appl., 448:2 (2017), 864–884
- B. P. Belinskiy, T. V. Savina, “The Schwarz reflection principle for harmonic functions in $mathbb R^2$ subject to the Robin condition”, J. Math. Anal. Appl., 348:2 (2008), 685–691
- F. Y. Maeda, N. Suzuki, “The integrability of superharmonic functions on Lipschitz domains”, Bull. London Math. Soc., 21:3 (1989), 270–278
- K. Miller, “Extremal barriers on cones with Phragmèn–Lindelöf theorems and other applications”, Ann. Mat. Pura Appl. (4), 90 (1971), 297–329
- J. Mateu, J. Orobitg, “Lipschitz approximation by harmonic functions and some applications to spectral sinthesis”, Indiana Univ. Math. J., 39:3 (1990), 703–736
- J. Verdera, “$C^m$-approximation by solutions of elliptic equations, and Calderon–Zygmund operators”, Duke Math. J., 55:1 (1987), 157–187
- П. В. Парамонов, “$C^m$-приближения гармоническими полиномами на компактных множествах в $mathbb{R}^n$”, Матем. сб., 184:2 (1993), 105–128
- М. Я. Мазалов, П. В. Парамонов, К. Ю. Федоровский, “Условия $C^m$-приближаемости функций решениями эллиптических уравнений”, УМН, 67:6(408) (2012), 53–100
Supplementary files
