Geodesic
On Smooth Functions That Are Even on the Boundary of a Ball
Informatics and Automation, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 156-161 
S. E. Zhukovskiy; K. V. Storozhuk. On Smooth Functions That Are Even on the Boundary of a Ball. Informatics and Automation, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 156-161. http://geodesic.mathdoc.fr/item/TRSPY_2023_321_a9/
@article{TRSPY_2023_321_a9,
     author = {S. E. Zhukovskiy and K. V. Storozhuk},
     title = {On {Smooth} {Functions} {That} {Are} {Even} on the {Boundary} of a {Ball}},
     journal = {Informatics and Automation},
     pages = {156--161},
     year = {2023},
     volume = {321},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2023_321_a9/}
}
TY  - JOUR
AU  - S. E. Zhukovskiy
AU  - K. V. Storozhuk
TI  - On Smooth Functions That Are Even on the Boundary of a Ball
JO  - Informatics and Automation
PY  - 2023
SP  - 156
EP  - 161
VL  - 321
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2023_321_a9/
LA  - ru
ID  - TRSPY_2023_321_a9
ER  - 
%0 Journal Article
%A S. E. Zhukovskiy
%A K. V. Storozhuk
%T On Smooth Functions That Are Even on the Boundary of a Ball
%J Informatics and Automation
%D 2023
%P 156-161
%V 321
%U http://geodesic.mathdoc.fr/item/TRSPY_2023_321_a9/
%G ru
%F TRSPY_2023_321_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] M. A. Krasnosel'skiy, A. I. Perov, A. I. Povolotskiy, and P. P. Zabreiko, Plane Vector Fields, Academic Press, New York, 1966 | MR | Zbl

[2] Zabreiko P.P., Krivko-Krasko A.V., “Obobschenie teorem Rollya i Darbu dlya funktsii dvukh peremennykh”, Vestsi Nats. akad. navuk Belarusi. Ser. fiz.-mat. navuk, 54:3 (2018), 290–299 | MR

[3] E. A. Kudryavtseva, “Realization of smooth functions on surfaces as height functions”, Sb. Math., 190:3 (1999), 349–405 | DOI | DOI | MR | Zbl

[4] Dombrowski P., “On global solutions for partial differential equations of first order”, Bull. Amer. Math. Soc., 67 (1961), 202–205 | DOI | MR | Zbl

[5] Francis G.K., “The folded ribbon theorem. A contribution to the study of immersed circles”, Trans. Amer. Math. Soc., 141 (1969), 271–303 | DOI | MR | Zbl