Reilly-type inequality for the Φ-Laplace operator on semislant submanifolds of Sasakian space forms
Yanlin Li, Fatemah Mofarreh, Ravi P. Agrawal, Akram Ali
Abstract
Abstract This paper aims to establish new upper bounds for the first positive eigenvalue of the Φ-Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the Φ-Laplacian operator on closed oriented m -dimensional semislant submanifolds in a Sasakian space form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>M</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mo>)</mml:mo> </mml:math> is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the Φ-Laplacian on semislant submanifolds in a sphere <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> with $\epsilon =1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> and $\Phi =2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> .