Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues

On Principal Eigenvalues for Periodic Parabolic Steklov Problems

LetΩ be aC2+γ domain in RN ,N ≥ 2, 0 < γ < 1. LetT>0 and let L be a uniformly parabolic operator Lu= ∂u/∂t−∑i, j(∂/∂xi)(ai j(∂u/∂xj)) +∑ j b j(∂u/∂xi) + a0u, a0 ≥ 0, whose coefficients, depending on (x, t) ∈Ω×R, are T periodic in t and satisfy some regularity assumptions. Let A be the N ×N matrix whose i, j entry is ai j and let ν be the unit exterior normal to ∂Ω. Let m be a T-periodic functio...

Nordhaus-Gaddum Type Inequalities for Laplacian and Signless Laplacian Eigenvalues

Let G be a graph with n vertices. We denote the largest signless Laplacian eigenvalue of G by q1(G) and Laplacian eigenvalues of G by μ1(G) > · · · > μn−1(G) > μn(G) = 0. It is a conjecture on Laplacian spread of graphs that μ1(G)−μn−1(G) 6 n − 1 or equivalently μ1(G) + μ1(G) 6 2n − 1. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph G, μ1(G)μ1(G) 6 n(n − ...

Payne-polya-weinberger Type Inequalities for Eigenvalues of Nonelliptic Operators

Let denote the Laplacian in the Euclidean space. The classic upper estimates, independent of the domain, for the gaps of eigenvalues of − , (− )2 and (− )k(k ≥ 3) were studied extensively by many mathematicians, cf. Payne, Polya and Weinberger [16], Hile and Yeh [10], Chen and Qian [2], Guo [8] etc.. The asymptotic behaviors of eigenvalues for degenerate elliptic operators were considered by Be...

Sloshing, Steklov and corners I: Asymptotics of sloshing eigenvalues

This is the first in a series of two papers aiming to establish sharp spectral asymptotics for Steklov type problems on planar domains with corners. In the present paper we focus on the two-dimensional sloshing problem, which is a mixed Steklov-Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We pr...

Isoperimetric Inequalities and Eigenvalues