On maximizing the fundamental frequency of the complement of an obstacle

Let $\mathrm{\Omega }?{\mathbb{R}}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let D be an arbitrary bounded domain referred to as an “obstacle”. We are interested in the behavior of the first Dirichlet eigenvalue ${\lambda }_1(\mathrm{\Omega }?(x+D))$.First, we prove an upper bound on ${\lambda }_1(\mathrm{\Omega }?(x+D))$ in terms of the distance of the set $x+D$ to the set of maximum points $x_0$ of the first Dirichlet ground state ${?}_{{\lambda }_1}>0$ of Ω. In short, a direct corollary is that if

(1)${\mu }_{\mathrm{\Omega }}:=\underset{x}{\max }?{\lambda }_1(\mathrm{\Omega }?(x+D))$

is large enough in terms of ${\lambda }_1(\mathrm{\Omega })$, then all maximizer sets $x+D$ of ${\mu }_{\mathrm{\Omega }}$ are close to each maximum point $x_0$ of ${?}_{{\lambda }_1}$.Second, we discuss the distribution of ${?}_{{\lambda }_1(\mathrm{\Omega })}$ and the possibility to inscribe wavelength balls at a given point in Ω.Finally, we specify our observations to convex obstacles D and show that if ${\mu }_{\mathrm{\Omega }}$ is sufficiently large with respect to ${\lambda }_1(\mathrm{\Omega })$, then all maximizers $x+D$ of ${\mu }_{\mathrm{\Omega }}$ contain all maximum points $x_0$ of ${?}_{{\lambda }_1(\mathrm{\Omega })}$.     

A Hardy type inequality for W02,1(Ω) functions

We consider functions $u\in W_0^{2,1}(\Omega )$, where $\Omega ?{\mathbb{R}}^N$ is a smooth bounded domain. We prove that $\frac{u(x)}{d(x)}\in W_0^{1,1}(\Omega )$ with$6\mathrm{?}\left(\frac{u(x)}{d(x)}\right)6_{L^1(\Omega )}?C6u{6}_{W^{2,1}(\Omega )},$ where d is a smooth positive function which coincides with $\mathrm{dist}(x,?\Omega )$ near ?Ω and C is a constant depending only on d and Ω.     

Positive solution for system of nonlinear first-order PBVPs on time scales

In this paper we are concerned with the following system of nonlinear first-order periodic boundary value problems on time scale $\mathbb{T}$where $f_i:[0,T]\times [0,+\infty {)}^n\to \mathbb{R}$ is continuous and there exists a constant $M_i>0$ such that$M_i{x}_i-f_i(t,x_1,x_2,\dots ,x_n)⩾0\text{for}(x_1,x_2,\dots ,x_n)\in [0,+\infty {)}^n,t\in [0,T]\text{.}$Some existence criteria of positive solution are established by using a fixed point theorem for operators on cone.     

The nonlinear complexity of level sequences over Z/(4)

For any sequence $\underline{a}$ over $\mathrm{Z}/(2^2)$, there is an unique 2-adic expansion $\underline{a}={\underline{a}}_0+{\underline{a}}_1·2$, where ${\underline{a}}_0$ and ${\underline{a}}_1$ are sequences over $\{0,1\}$ and can be regarded as sequences over the binary field $\mathit{GF}(2)$ naturally. We call ${\underline{a}}_0$ and ${\underline{a}}_1$ the level sequences of $\underline{a}$. Let $f(x)$ be a primitive polynomial of degree $n$ over $\mathrm{Z}/(2^2)$, and $\underline{a}$ be a primitive sequence generated by $f(x)$. In this paper, we discuss how many bits of ${\underline{a}}_1$ can determine uniquely the original primitive sequence $\underline{a}$. This issue is equivalent with one to estimate the whole nonlinear complexity, $\mathit{NL}(f(x),2^2)$, of all level sequences of $f(x)$. We prove that $4n$ is a tight upper bound of $\mathit{NL}(f(x),2^2)$ if $f(x)(\mathrm{mod}2)$ is a primitive trinomial over $\mathit{GF}(2)$. Moreover, the experimental result shows that $\mathit{NL}(f(x),2^2)$ varies around $4n$ if $f(x)(\mathrm{mod}2)$ is a primitive polynomial over $\mathit{GF}(2)$. From this result, we can deduce that $\mathit{NL}(f(x),2^2)$ is much smaller than $L(f(x),2^2)$, where $L(f(x),2^2)$ is the linear complexity of level sequences of $f(x)$.