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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Volume 321, Pages 156–161
DOI: https://doi.org/10.4213/tm4319 (Mi tm4319) 		 
On Smooth Functions That Are Even on the Boundary of a Ball

S. E. Zhukovskiya, K. V. Storozhukb

a V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, ul. Profsoyuznaya 65, Moscow, 117997 Russia
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
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DOI: https://doi.org/10.4213/tm4319
Abstract: We show how to construct a smooth function without critical points on the ball $B^n$, $n>1$, that is even on its boundary $S^{n-1}$. In particular, it follows that the corresponding generalization of Rolle's theorem to dimensions $n>1$ does not hold.
Funding agency Grant number
Russian Science Foundation 22-11-00042
Ministry of Science and Higher Education of the Russian Federation FWNF-2022-0006
The work of the first author was supported by the Russian Science Foundation under grant no. 22-11-00042, https://rscf.ru/en/project/22-11-00042/, and performed at the V. A. Trapeznikov Institute of Control Sciences, RAS. The work of the second author was performed within the framework of the state assignment of the Sobolev Institute of Mathematics, SB RAS, project no. FWNF-2022-0006. Sections 2 and 6 were written by the first author; Sections 1, 3, 4, and 5 were written by the second author. All results in this paper are products of the authors' collaborative work.
Received: March 28, 2022
Revised: July 10, 2022
Accepted: January 9, 2023
English version:
Proceedings of the Steklov Institute of Mathematics, 2023, Volume 321, Pages 143–148
DOI: https://doi.org/10.1134/S0081543823020104
Bibliographic databases:
Document Type: Article
UDC: 517.27
Language: Russian
Citation: S. E. Zhukovskiy, K. V. Storozhuk, “On Smooth Functions That Are Even on the Boundary of a Ball”, Optimal Control and Dynamical Systems, Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze, Trudy Mat. Inst. Steklova, 321, Steklov Math. Inst., Moscow, 2023, 156–161; Proc. Steklov Inst. Math., 321 (2023), 143–148
Citation in format AMSBIB
\Bibitem{ZhuSto23}
\by S.~E.~Zhukovskiy, K.~V.~Storozhuk
\paper On Smooth Functions That Are Even on the Boundary of a Ball
\inbook Optimal Control and Dynamical Systems
\bookinfo Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze
\serial Trudy Mat. Inst. Steklova
\yr 2023
\vol 321
\pages 156--161
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4319}
\crossref{https://doi.org/10.4213/tm4319}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4643638}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2023
\vol 321
\pages 143--148
\crossref{https://doi.org/10.1134/S0081543823020104}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85171193391}
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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