In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack’s inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting properties of Schrodinger equation are derived. Mathematics Subject Classific...

In this paper, we study a Fokker-Planck equation of the form ut = I[u] + div(xu) where the operator I, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the only steady state of the associated stationnary equation.

The existence of periodic solutions of a higher order nonlinear functional difference equation with p-Laplacian is studied. Sufficient conditions for the existence of periodic solutions of such equation are established. The result is based on Mawhin′s continuation theorem. The methods used to estimate the priori bound on periodic solutions are very technical.

We characterize well-posedness in Hölder spaces for an abstract version of the equation (∗) u′′ + λu′′′ = c(∆u + μ∆u′) + f which model the vibrations of flexible structures possessing internal material damping and external force f . As a consequence, we show that in case of the Laplacian with Dirichlet boundary conditions, equation (*) is always well-posed provided 0 < λ < μ.

In this paper, we study a Fokker-Planck equation of the form ut = I[u] + div(xu) where the operator I, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the only steady state of the associated stationnary equation.

The family G of connected graphs with second largest Laplacian eigenvalue at most θ, where θ = 3.2470 is the largest root of the equation μ−5μ+6μ−1 = 0, is characterized by Wu, Yu and Shu [Y.R. Wu, G.L. Yu and J.L. Shu, Graphs with small second largest Laplacian eigenvalue, European J. Combin. 36 (2014) 190–197]. Let G(a, b, c, d) be a graph with order n = 2a + b + 2c + 3d + 1 that consists of ...

Recently, one debate in the literature is whether the fractional Schródinger equation in an infinite potential well has the same eigenfunctions as those of its standard (non-fractional) counterpart. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schródinger equation analytically. In this talk, we numerically stu...

We study the spectrum of the scalar Laplacian on the five-dimensional toric Sasaki-Einstein manifolds Y . The eigenvalue equation reduces to Heun’s equation, which is a Fuchsian equation with four regular singularities. We show that the ground states, which are given by constant solutions of Heun’s equation, are identified with BPS states corresponding to the chiral primary operators in the dua...

The paper studies boundary integral operators of the bi{Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving mixed boundary value problems for the biharmonic equation on a nonsmooth plane domain, analyse the solvability of the co...

Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Abstract We obtain solutions of the nonlinear degenerate parabolic equation ∂ ρ ∂ t = div ρ ∇c ⋆ [ ∇ (F ′ (ρ) + V) ] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires less uniform convexity assumption than that imposed by Alt and ...