Eigenvalues of the Laplacian with Neumann boundary conditions

Lyapunov Inequalities for Neumann Boundary Conditions at Higher Eigenvalues

This paper is devoted to the study of Lyapunov-type inequality for Neumann boundary conditions at higher eigenvalues. Our main result is derived from a detailed analysis about the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the use of suitable minimization problems. This method of proof allows to obtain new information on Lyapunov constant...

On the Positive and Negative Solutions of Laplacian Bvp with Neumann Boundary Conditions

In this paper, we consider the following Neumann boundary value problem { −u′′(x) = u(x)− λ|u(x)|, x ∈ (0, 1), u′(0) = 0 = u′(1), where λ ∈ R is parameter. We study the positive and negative solutions of this problem with respect to a parameter ρ (i.e. u(0) = ρ) in all R∗. By using a quadrature method, we obtain our results. Also we provide some details about the solutions that are obtained.

Eigenvalues of the Laplacian on domains with fractal boundary

Consider the Laplacian operator on a bounded open domain in Euclidean space with Dirichlet boundary conditions. We show that for each number D with 1 < D < 2, there are two bounded open domains in R2 of the same area, with their boundaries having Minkowski dimension D, and having the same content, yet the secondary terms for the eigenvalue counts are not the same. This was shown earlier by Lapi...

LOCAL AND NONLOCAL WEIGHTED p-LAPLACIAN EVOLUTION EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

Blow-up phenomena for p-Laplacian parabolic problems with Neumann boundary conditions

where p > 2 and is a bounded domain in Rn (n≥ 2) with smooth boundary ∂ . By introducing some appropriate auxiliary functions and technically using maximum principles, we establish conditions to guarantee that the solution blows up in some finite time or remains global. In addition, the upper estimates of blow-up rate and global solution are specified. We also obtain an upper bound of blow-up t...