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.
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.
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
\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}