On solvability of one nonlocal boundary problem for the Laplace operator with opposite flows at the part of the boundary

2016 ◽  
Author(s):  
Issabek Orazov ◽  
Gani A. Besbaev
Keyword(s):  
Boundary Problem ◽  
Laplace Operator ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Problem ◽  
The Laplace Operator

scholarly journals ONE DIMENSIONAL BROWNIAN MOTION WITH HOLDING AND JUMPING BOUNDARY

2022 ◽  
Author(s):  
Yuk Leung
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Probability Density ◽  
Boundary Problem ◽  
Exponential Distribution ◽  
Boundary Point ◽  
Nonlocal Boundary ◽  
Unit Interval ◽  
One Dimensional ◽  
Nonlocal Boundary Problem

Let a particle start at some point in the unit interval I := [0, 1] and undergo Brownian motion in I until it hits one of the end points. At this instant the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside I with a probability density μ0 or μ1 parametrized by the boundary point it was from. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper [10]. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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