On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.     

Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry

Let $n\in \mathbb{N}$ with $n?2$, $a\in (?1,0)\cup (0,1]$ and $f:(0,1)\times (0,\infty )\to \mathbb{R}$ such that for each $u\in (0,\infty )$, $r?{(1+a{r}^2)}^{(n+2)/2}f(r,{(1+a{r}^2)}^{?(n?2)/2}u):(0,1)\to \mathbb{R}$ is nonincreasing. We show that each positive solution of$\mathrm{\Delta }u+f(|x|,u)=0\text{in}B,u=0\text{on}?B$ is radially symmetric, where B is the open unit ball in ${\mathbb{R}}^N$.     

Degree Sequence of Tight Distance Graphs

A graph G on n vertices is a tight distance graph if there exists a set $D\subseteq \{1,2,\dots ,n-1\}$ such that $V(G)=\{0,1,\dots ,n-1\}$ and $ij\in E(G)$ if and only if $|i-j|\in D$. A characterization of the degree sequences of tight distance graphs is given. This characterization yields a fast method for recognizing and realizing degree sequences of tight distance graphs.     

The Lp Robin problem for Laplace equations in Lipschitz and (semi-)convex domains

Let $n\ge 3$ and Ω be a bounded Lipschitz domain in ${\mathbb{R}}^n$. Assume that $p\in (2,\infty )$ and the function $b\in L^{\infty }(?\mathrm{\Omega })$ is non-negative, where ?Ω denotes the boundary of Ω. Denote by ν the outward unit normal to ?Ω. In this article, the authors give two necessary and sufficient conditions for the unique solvability of the Robin problem for the Laplace equation $\mathrm{\Delta }u=0$ in Ω with boundary data $?u/?\nu +bu=f\in L^p(?\mathrm{\Omega })$, respectively, in terms of a weak reverse Hölder inequality with exponent p or the unique solvability of the Robin problem with boundary data in some weighted $L^2(?\mathrm{\Omega })$ space. As applications, the authors obtain the unique solvability of the Robin problem for the Laplace equation in the bounded (semi-)convex domain Ω with boundary data in (weighted) $L^p(?\mathrm{\Omega })$ for any given $p\in (1,\infty )$.     

On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

On the longest block in Lüroth expansion

In this paper, for a finite subset $A?\{2,3,?\}$, we introduce the notion of longest block function $L_n(x,A)$ for the Lüroth expansion of $x\in [0,1)$ with respect to A and consider the asymptotic behavior of $L_n(x,A)$ as n tends to ∞. We also obtain the Hausdorff dimensions of the level sets and exceptional set arising from the longest block function.     

Positive effects of repulsion on boundedness in a fully parabolic attraction–repulsion chemotaxis system with logistic source

In this paper we study the global boundedness of solutions to the fully parabolic attraction–repulsion chemotaxis system with logistic source: $u_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)+\xi \mathrm{?}?(u\mathrm{?}w)+f(u)$, $v_t=\mathrm{\Delta }v?\beta v+\alpha u$, $w_t=\mathrm{\Delta }w?\delta w+\gamma u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$), where χ, α, ξ, γ, β and δ are positive constants, and $f:\mathbb{R}\to \mathbb{R}$ is a smooth function generalizing the logistic source $f(s)=a?b{s}^{\theta }$ for all $s\ge 0$ with $a\ge 0$, $b>0$ and $\theta \ge 1$. It is shown that when the repulsion cancels the attraction (i.e. $\chi \alpha =\xi \gamma$), the solution is globally bounded if $n\le 3$, or $\theta >{\theta }_n:=\min ?\{\frac{n+2}{4},\frac{n\sqrt{n^2+6n+17}?n^2?3n+4}{4}\}$ with $n\ge 2$. Therefore, due to the inhibition of repulsion to the attraction, in any spatial dimension, the exponent θ is allowed to take values less than 2 such that the solution is uniformly bounded in time.     

Level sets of β-expansions

Let ${\{{\mathrm{\epsilon }}_n(x)\}}_{n?1}$ be the sequence of β-digits of a real number $x\in (0\text{,}1)$, with the golden number $\beta =(\sqrt{5}+1)/2$ as basis. For any $0?p?1/2$, any $0<\tau ?1$ and any real number a, we consider the level set consisting of numbers x such that ${\sum }_{n=1}^{\infty }({\mathrm{\epsilon }}_n(x)?p)/n^{\tau }=a$. We prove that the Hausdorff dimension of this set is independent of a and τ, and that it is equal to $\log f(p)/\log \beta$ where $f(p)={(1?p)}^{1?p}/({(1?2p)}^{1?2p}{p}^p)$. To cite this article: A. Fan, H. Zhu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).