Diffusion versus absorption in semilinear parabolic problems

We study the limit, when $k\to \infty$, of the solutions $u=u_k$ of (E) ${?}_{t}u?\mathrm{\Delta }u+h(t)u^q=0$ in ${\mathbb{R}}^N\times (0,\infty )$, $u_k(?,0)=k{\delta }_0$, with $q>1$, $h(t)>0$. If $h(t)={\mathrm{e}}^{?\omega (t)/t}$ where $\omega >0$ satisfies to ${\int }_0^1\sqrt{\omega (t)}{t}^{\mathrm{?1}}\mathrm{d}t<\infty$, the limit function $u_{\infty }$ is a solution of (E) with a single singularity at $(0,0)$, while if $\omega (t)\equiv 1$, $u_{\infty }$ is the maximal solution of (E). We examine similar questions for equations such as ${?}_{t}u?\mathrm{\Delta }{u}^m+h(t)u^q=0$ with $m>1$ and ${?}_{t}u?\mathrm{\Delta }u+h(t){\mathrm{e}}^u=0$. To cite this article: A. Shishkov, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Factorization of a class of polynomials over finite fields

We study the factorization of polynomials of the form $F_r(x)=b{x}^{q^r+1}?a{x}^{q^r}+dx?c$ over the finite field ${\mathbb{F}}_q$. We show that these polynomials are closely related to a natural action of the projective linear group $\mathrm{PGL}(2,q)$ on non-linear irreducible polynomials over ${\mathbb{F}}_q$. Namely, irreducible factors of $F_r(x)$ are exactly those polynomials that are invariant under the action of some non-trivial element $[A]\in \mathrm{PGL}(2,q)$. This connection enables us to enumerate irreducibles which are invariant under $[A]$. Since the class of polynomials $F_r(x)$ includes some interesting polynomials like $x^{q^r}?x$ or $x^{q^r+1}?1$, our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over ${\mathbb{F}}_q$. At the same time, we generalize recent results about certain invariant polynomials over the binary field ${\mathbb{F}}_2$.     

Inviscid limit for the derivative Ginzburg–Landau equation with small data in modulation and Sobolev spaces

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation $u_t=(\nu +\mathrm{i})\mathrm{\Delta }u+{\overrightarrow{\lambda }}_1?\mathrm{?}({|u|}^{2}u)+({\overrightarrow{\lambda }}_2?\mathrm{?}u){|u|}^2+\alpha {|u|}^{2\delta }u$, where $\delta \in \mathbb{N}$, ${\overrightarrow{\lambda }}_1,{\overrightarrow{\lambda }}_2$ are complex constant vectors, $\nu \in [0,1]$, $\alpha \in \mathbb{C}$. For $n\ge 3$, we show that it is uniformly global well posed for all $\nu \in [0,1]$ if initial data $u_0$ in modulation space $M_{2,1}^s$ and Sobolev spaces $H^{s+n/2}$ ($s>3$) and $6u_0{6}_{L^2}$ is small enough. Moreover, we show that its solution will converge to that of the derivative Schrödinger equation in $C(0,T;L^2)$ if $\nu \to 0$ and $u_0$ in $M_{2,1}^s$ or $H^{s+n/2}$ with $s>4$. For $n=2$, we obtain the local well-posedness results and inviscid limit with the Cauchy data in $M_{1,1}^s$ ($s>3$) and $6u_0{6}_{L^1}?1$.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

Recovery of sparsest signals via ℓq-minimization

In this paper, it is proved that every s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ can be exactly recovered from the measurement vector $\mathbf{z}=\mathbf{Ax}\in {\mathbb{R}}^m$ via some ${?}^q$-minimization with $0<q?1$, as soon as each s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ is uniquely determined by the measurement z. Moreover it is shown that the exponent q in the ${?}^q$-minimization can be so chosen to be about $0.6796\times (1?{\delta }_{2s}(\mathbf{A}))$, where ${\delta }_{2s}(\mathbf{A})$ is the restricted isometry constant of order 2s for the measurement matrix A.     

A note on plane pointless curves

Let $d(q)$ denote the minimal degree of a smooth projective plane curve that is defined over the finite field ${\mathbb{F}}_q$ and does not contain ${\mathbb{F}}_q$ rational points. We are interested in the asymptotic behavior of $d(q)$ for $q\to \infty$. To the best of the author's knowledge the problem of estimating the asymptotic behavior of $d(q)$ was not considered previously. In this note we establish the following bounds:(1)$\frac{1}{4}⩽\underset{q\to \infty }{\underline{\lim }}{\log }_{q}d(q)⩽\frac{1}{3}.$ More specifically, for every characteristic $p>3$ we construct a sequence of pointless Fermat curves$x^{d_k}+y^{d_k}+z^{d_k}=0,\text{over }{\mathbb{F}}_{p^{m_k}},$ such that ${\lim }_{k\to \infty }{\log }_{p^{m_k}}{d}_k=1/3$.     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

On the geography of threefolds of general type

Let X be a complex nonsingular projective 3-fold of general type. We show that there are positive constants c, $c^{\prime }$ and $m_1$ such that $\chi ({\omega }_X)??c\mathrm{Vol}(X)$ and $P_m(X)?c^{\prime }{m}^3\mathrm{Vol}(X)$ for all $m?m_1$.     

A Gauss sum estimate in arbitrary finite fields

We establish bounds on exponential sums ${\sum }_{x\in {\mathbb{F}}_q}\psi (x^n)$ where $q=p^m$, p prime, and ψ an additive character on ${\mathbb{F}}_q$. They extend the earlier work of Bourgain, Glibichuk, and Konyagin to fields that are not of prime order $(m?2)$. More precisely, a non-trivial estimate is obtained provided n satisfies $\gcd (n,\frac{q?1}{p^{\nu }?1})<p^{?\nu }{q}^{1?\epsilon }$ for all $1?\nu <m$, $\nu |m$, where $\epsilon >0$ is arbitrary. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Optimal Wloc2,2-regularity,Pohozhaevʼs identity,and nonexistence of weak solutions to some quasilinear elliptic equations

We begin by establishing a sharp (optimal) $W_{\mathrm{loc}}^{2,2}$-regularity result for bounded weak solutions to a nonlinear elliptic equation with the p-Laplacian, ${\mathrm{\Delta }}_{p}u\stackrel{\mathrm{def}}{=}\mathrm{div}({|\mathrm{?}u|}^{p?2}\mathrm{?}u)$, $1<p<\infty$. We develop very precise, optimal regularity estimates on the ellipticity of this degenerate (for $2<p<\infty$) or singular (for $1<p<2$) problem. We apply this regularity result to prove Pohozhaev?s identity for a weak solution $u\in W^{1,p}(\Omega )$ of the elliptic Neumann problem(P)$?{\mathrm{\Delta }}_{p}u+W^{\prime }(u)=f(x)\text{in}\Omega ;?u/?\nu =0\text{on}?\Omega .$ Here, Ω is a bounded domain in ${\mathbb{R}}^N$ whose boundary ?Ω is a $C^2$-manifold, $\nu \equiv \nu (x_0)$ denotes the outer unit normal to ?Ω at $x_0\in ?\Omega$, $x=(x_1,\dots ,x_N)$ is a generic point in Ω, and $f\in L^{\infty }(\Omega )\cap W^{1,1}(\Omega )$. The potential $W:\mathbb{R}\to \mathbb{R}$ is assumed to be of class $C^1$ and of the typical double-well shape of type $W(s)={|1?{|s|}^{\beta }|}^{\alpha }$ for $s\in \mathbb{R}$, where $\alpha ,\beta >1$ are some constants. Finally, we take an advantage of the Pohozhaev identity to show that problem (P) with $f\equiv 0$ in Ω has no phase transition solution $u\in W^{1,p}(\Omega )$ ($1<p?N$), such that $?1?u?1$ in Ω with $u\equiv ?1$ in ${\Omega }_{?1}$ and $u\equiv 1$ in ${\Omega }_1$, where both ${\Omega }_{?1}$ and ${\Omega }_1$ are some nonempty subdomains of Ω. Such a scenario for u is possible only if $N=1$ and ${\Omega }_{?1}$, ${\Omega }_1$ are finite unions of suitable subintervals of the open interval $\Omega ?{\mathbb{R}}^1$.     

Existence and multiplicity of solutions for a discontinuous problems with critical Sobolev exponents

In this paper, we consider the equation$?{\mathrm{\Delta }}_{p}u=\lambda {|u|}^{p^{?}?2}u+f(x,u)\text{in}{\mathbf{R}}^N,$ with discontinuous nonlinearity, where $1<p<N$, $\lambda >0$ is a real parameter and $p^{?}=\frac{Np}{N?p}$ is the critical Sobolev exponent. Under proper conditions on f, applying the nonsmooth critical point theory for locally Lipschitz functionals, we obtain at least one nontrivial nonnegative solution provided that $\lambda <{\lambda }_0$ and for any $k\in \mathbf{N}$, it has k pairs of nontrivial solutions if $\lambda <{\lambda }_k$, where ${\lambda }_0$ and ${\lambda }_k$ are positive numbers. In particular, we obtain the existence results for f is discontinuous in just one point.