scholarly journals On the lower bound of the principal eigenvalue of a nonlinear operator

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Yucheng Tu
Keyword(s):  
Lower Bound ◽  
Neumann Boundary Condition ◽  
Measure Space ◽  
Nonlinear Operator ◽  
Principal Eigenvalue ◽  
Neumann Boundary ◽  
Diffusion Operator ◽  
Smooth Metric Measure Space ◽  
Partial Differ ◽  
Sharp Lower Bound

AbstractWe prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator $$L_p$$ L p on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying $$BE(\kappa ,N)$$ B E ( κ , N ) for $$\kappa \ne 0$$ κ ≠ 0 . Our results extends the work of Koerber Valtorta (Calc Vari Partial Differ Equ. 57(2), 49 2018) for case $$\kappa =0$$ κ = 0 and Naber–Valtorta (Math Z 277(3–4):867–891, 2014) for the p-Laplacian.

scholarly journals Eigenvalues of the drifted Laplacian on complete metric measure spaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Xu Cheng ◽  
Detang Zhou
Keyword(s):  
Lower Bound ◽  
Measure Space ◽  
Logarithmic Sobolev Inequality ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Dimensional Manifold ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Space ◽  
Measure Spaces ◽  
Multiplicity Formula

In this paper, first we study a complete smooth metric measure space [Formula: see text] with the ([Formula: see text])-Bakry–Émery Ricci curvature [Formula: see text] for some positive constant [Formula: see text]. It is known that the spectrum of the drifted Laplacian [Formula: see text] for [Formula: see text] is discrete and the first nonzero eigenvalue of [Formula: see text] has lower bound [Formula: see text]. We prove that if the lower bound [Formula: see text] is achieved with multiplicity [Formula: see text], then [Formula: see text], [Formula: see text] is isometric to [Formula: see text] for some complete [Formula: see text]-dimensional manifold [Formula: see text] and by passing an isometry, [Formula: see text] must split off a gradient shrinking Ricci soliton [Formula: see text], [Formula: see text]. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian [Formula: see text] for properly immersed self-shrinkers in the Euclidean space [Formula: see text], [Formula: see text] and show the discreteness of the spectrum of [Formula: see text] and a logarithmic Sobolev inequality.

Minimization of eigenvalues for the Camassa–Holm equation

2020 ◽  
pp. 2050021
Author(s):  
Hao Feng ◽  
Gang Meng
Keyword(s):  
Boundary Condition ◽  
Lower Bound ◽  
Minimization Problem ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Smallest Eigenvalue ◽  
Holm Equation ◽  
The One ◽  
The Relationship ◽  
Optimal Lower Bound

A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem [Formula: see text] We will study in this paper the optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation with the Neumann boundary condition when the [Formula: see text] norm of potentials is given. First, we will study the optimal lower bound for the smallest eigenvalue in the measure differential equations to make our results more applicable. Second, Based on the relationship between the minimization problem of the smallest eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

scholarly journals Compactness and Sharp Lower Bound for a 2D Smectics Model

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Michael Novack ◽  
Xiaodong Yan
Keyword(s):  
Lower Bound ◽  
Sharp Lower Bound
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