Geodesic
First $p$-Steklov eigenvalue under geodesic curvature flow
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 1, pp. 293-306
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the first nonzero $p$-Steklov eigenvalue on a two-dimensional compact Riemannian manifold with a smooth boundary along the geodesic curvature flow. We prove that the first nonzero $p$-Steklov eigenvalue is nondecreasing if the initial metric has positive geodesic curvature on boundary $\partial M$ and Gaussian curvature is identically equal to zero in $M$ along the un-normalized geodesic curvature flow. An eigenvalue estimation is also obtained along the normalized geodesic curvature flow.
Keywords: geodesic curvature, geodesic curvature flow.
Mots-clés : $p$-Steklov eigenvalue 
@article{SEMR_2024_21_1_a17,
     author = {A. Saha and S. Azami and S. K. Hui},
     title = {First $p${-Steklov} eigenvalue under geodesic curvature flow},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {293--306},
     year = {2024},
     volume = {21},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2024_21_1_a17/}
}
TY  - JOUR
AU  - A. Saha
AU  - S. Azami
AU  - S. K. Hui
TI  - First $p$-Steklov eigenvalue under geodesic curvature flow
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2024
SP  - 293
EP  - 306
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SEMR_2024_21_1_a17/
LA  - en
ID  - SEMR_2024_21_1_a17
ER  - 
%0 Journal Article
%A A. Saha
%A S. Azami
%A S. K. Hui
%T First $p$-Steklov eigenvalue under geodesic curvature flow
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2024
%P 293-306
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/SEMR_2024_21_1_a17/
%G en
%F SEMR_2024_21_1_a17
A. Saha; S. Azami; S. K. Hui. First $p$-Steklov eigenvalue under geodesic curvature flow. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 1, pp. 293-306. http://geodesic.mathdoc.fr/item/SEMR_2024_21_1_a17/

[1] S. Azami, “Evolution of eigenvalue of the Wentzell-Laplace operator along the geodesic curvature flow”, Indian J. Pure Appl. Math., 2023 | DOI | MR

[2] S. Brendle, “A family of curvature flows on surfaces with boundary”, Math. Z., 241:4 (2002), 829–869 | DOI | MR | Zbl

[3] S. Brendle, “Curvature flows on surfaces with boundary”, Math. Ann., 324:3 (2002), 491–519 | DOI | MR | Zbl

[4] X. Cao, “Eigenvalues of $(-\Delta + \frac{R}{2})$ on manifolds with nonnegative curvature operator”, Math. Ann., 337:2 (2007), 435–441 | DOI | MR | Zbl

[5] X. Cao, “First eigenvalues of geometric operators under the Ricci flow”, Proc. Am. Math. Soc., 136:11 (2008), 4075–4078 | DOI | MR | Zbl

[6] J. Chen, Y. Wang, “Liouville type theorems for the p-harmonic functions on certain manifolds”, Pac. J. Math., 282:2 (2016), 313–327 | DOI | MR | Zbl

[7] P.T. Ho, “Prescribed curvature flow on surfaces”, Indiana Univ. Math. J., 60:5 (2011), 1527–1542 | DOI | MR

[8] P.T. Ho, “First eigenvalues of geometric operators under the Yamabe flow”, Ann. Global Anal. Geom., 54:4 (2018), 449–472 | DOI | MR | Zbl

[9] P.T. Ho, H. Koo, “Evolution of the Steklov eigenvalue under geodesic curvature flow”, Manuscr. Math., 159:3-4 (2019), 453–473 | DOI | MR | Zbl

[10] A. Jollivet, V. Sharafutdinov, “An estimate for the Steklov zeta function of a planar domain derived from a first variation formula”, J. Geom. Anal., 32:5 (2022), 161 | DOI | MR | Zbl

[11] J.J. Manfredi, M. Parviainen, J.D. Rossi, “On the definition and properties of $p$-harmonious functions”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 11:2 (2012), 215–241 | DOI | MR | Zbl

[12] B. Osgood, R. Phillips, P. Sarnak, “Extremals of determinants of Laplacians”, J. Funct. Anal., 80:1 (1988), 148–211 | DOI | MR | Zbl

[13] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159 | DOI | Zbl

[14] A. Saha, S. Azami, S.K. Hui, “First eigenvalue of weighted $p$-Laplacian under cotton flow”, Filomat, 35:9 (2021), 2919–2926 | DOI | MR

[15] A. Saha, S. Azami, S.K. Hui, “Eigenvalue of $(p, q)$-Laplace system along the forced mean curvature flow”, Iran. J. Sci. Technol. Trans. Sci., 45 (2021), 2041–2047 | DOI | MR

[16] A. Saha, S. Azami, S.K. Hui, “Eigenvalue of $(p,q)$-Laplace system along the powers of mean curvature flow”, Filomat, 36:18 (2022), 6269–6278 | DOI | MR

[17] J. Roth, “Extrinsic upper bounds the first eigenvalue of the p-Steklov problem on submanifolds”, Commun. Math., 30:1 (2022), 49–61 | DOI | MR

[18] J. Roth, A. Upadhyay, “Reilly-type upper bounds for the p-Steklov problem on submanifolds”, Bull. Aust. Math. Soc., 108:3 (2023), 492–503 | DOI | MR | Zbl

[19] H. Zhang, “Evolution of curvatures on a surface with boundary to prescribed functions”, Manuscr. Math., 149:1-2 (2016), 153–170 | DOI | MR | Zbl