Spectral gaps for water waves above a corrugated bottom

In this paper, the essential spectrum of the linear problem on water waves in a water layer and in a channel with a gently corrugated bottom is studied. We show that, under a certain geometric condition, the essential spectrum has spectral gaps. In other words, there exist intervals in the positive real semi-axis that are free of the spectrum but have their endpoints in it. The position and the length of the gaps are found out by applying an asymptotic analysis to the model problem in the periodicity cell

Operator Inequalities Associated with Jensen’s Inequality

Abstract. We give a survey of various operator inequalities associated with Jensen’s inequality and study the class of operator convex functions of several variables. Related questions are considered. 1 Jensen’s classical inequality Let I be a real interval. A function f: I → R is said to be convex, if f(λt+ (1 − λ)s) ≤ λf(t) + (1 − λ)f(s)(1.1) for all t, s ∈ I and every λ ∈ [0, 1]. Notice that the definition, in order to be meaningful, requires that f can be evaluated in λt+(1−λ)s, or equivalently that I is convex. But this is satisfied because the convex subsets of R are the intervals. If f satisfies (1.1) just for λ = 1/2, then f is said to be mid-point convex. It is easy to establish that a continuous and mid-point convex function is convex. The geometric interpretation of (1.1) is that the graph of f is below the chord and consequently above the extensions of the chord. This entails that a convex function defined on an open interval is continuous. Condition (1.1) can be reformulated a