scholarly journals Mean Curvature Type Flow with Perpendicular Neumann Boundary Condition inside a Convex Cone

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Fangcheng Guo ◽  
Guanghan Li ◽  
Chuanxi Wu

We investigate the evolution of hypersurfaces with perpendicular Neumann boundary condition under mean curvature type flow, where the boundary manifold is a convex cone. We find that the volume enclosed by the cone and the evolving hypersurface is invariant. By maximal principle, we prove that the solutions of this flow exist for all time and converge to some part of a sphere exponentially asttends to infinity.

2018 ◽  
Vol 62 (2) ◽  
pp. 459-469
Author(s):  
Ben Lambert

AbstractWe prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension n = 2. This then implies that the mean curvature flow exists for all time and converges to a translating solution.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Agil K. Khanmamedov ◽  
Nigar F. Gafarova

AbstractAn anharmonic oscillator {T(q)=-\frac{d^{2}}{dx^{2}}+x^{2}+q(x)} on the half-axis {0\leq x<\infty} with the Neumann boundary condition is considered. By means of transformation operators, the direct and inverse spectral problems are studied. We obtain the main integral equations of the inverse problem and prove that the main equation is uniquely solvable. An effective algorithm for reconstruction of perturbed potential is indicated.


2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.


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