The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$. These groups contain as special cases the odd dimensional general linear groups ${\mathrm{GL}}_{2n+1}(R)$ where R is any ring, the odd dimensional orthogonal and symplectic groups ${\mathrm{O}}_{2n+1}(R)$ and ${\mathrm{Sp}}_{2n+1}(R)$ where R is any commutative ring and further the first author's even dimensional unitary groups ${\mathrm{U}}_{2n}(R,\mathrm{\Lambda })$ where $(R,\mathrm{\Lambda })$ is any form ring. We classify the E-normal subgroups of the groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ (i.e. the subgroups which are normalized by the elementary subgroup ${\mathrm{EU}}_{2n+1}(R,\mathrm{\Delta })$), under the condition that R is either a semilocal or quasifinite ring with involution and $n\ge 3$. Further we investigate the action of ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ by conjugation on the set of all E-normal subgroups.     

Construction and stability of blowup solutions for a non-variational semilinear parabolic systemConstruction et stabilité de solutions explosives pour un système parabolique sémilinéaire non-variationel

We consider the following parabolic system whose nonlinearity has no gradient structure:

$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+|v{|}^{p?1}v,\hfill & {?}_{t}v=\mu \mathrm{\Delta }v+|u{|}^{q?1}u,\hfill \\ u(?,0)=u_0,\hfill & v(?,0)=v_0,\hfill \end{array}$

in the whole space ${\mathbb{R}}^N$, where $p,q>1$ and $\mu >0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0,v_0)$ simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:

$\{\begin{array}{cc}\hfill & u(x,t)～\mathrm{\Gamma }{[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(p+1)}{pq?1}},\hfill \\ \hfill & v(x,t)～\gamma {[(T?t)(1+\frac{b|x?a{|}^2}{(T?t)|\log ?(T?t)|})]}^{?\frac{(q+1)}{pq?1}},\hfill \end{array}$

with $b=b(p,q,\mu )>0$ and $(\mathrm{\Gamma },\gamma )=(\mathrm{\Gamma }(p,q),\gamma (p,q))$. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case $\mu =1$; and the fact that the case $\mu \ne 1$ breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.     

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 $p\in (1,\infty )$.     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.     

Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption

The chemotaxis system

$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\mathrm{?}?\left(\frac{u}{v}\mathrm{?}v\right),\hfill \\ v_t=\mathrm{\Delta }v?uv,\hfill \end{array}(?)$

is considered under homogeneous Neumann boundary conditions in the ball $\mathrm{\Omega }=B_R(0)?{\mathbb{R}}^n$, where $R>0$ and $n\ge 2$.Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data.For all suitably regular and radially symmetric initial data $(u_0,v_0)$ satisfying $u_0\ge 0$ and $v_0>0$, the present paper establishes the existence of a globally defined pair $(u,v)$ of radially symmetric functions which are continuous in $(\overline{\mathrm{\Omega }}?\{0\})\times [0,\infty )$ and smooth in $(\overline{\mathrm{\Omega }}?\{0\})\times (0,\infty )$, and which solve the corresponding initial-boundary value problem for (?) with $(u(?,0),v(?,0))=(u_0,v_0)$ in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (?) involving arbitrarily large initial data.     

Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant

$\{\begin{array}{cc}{u}_t=d_1\mathrm{\Delta }u?\mathrm{?}?(u{\chi }_1(w)\mathrm{?}w)+{\mu }_{1}u(1?u?a_{1}v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=d_2\mathrm{\Delta }v?\mathrm{?}?(v{\chi }_2(w)\mathrm{?}w)+{\mu }_{2}v(1?a_{2}u?v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=d_3\mathrm{\Delta }w?(\alpha u+\beta v)w,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$) with smooth boundary, where the initial data $(u_0,v_0)\in {(C^0(\stackrel{￣}{\mathrm{\Omega }}))}^2$ and $w_0\in W^{1,\infty }(\mathrm{\Omega })$ are non-negative and the parameters $d_1,d_2,d_3>0$, ${\mu }_1,{\mu }_2>0$, $a_1,a_2>0$ and $\alpha ,\beta >0$. The chemotactic function ${\chi }_i(w)$ ($i=1,2$) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for $i=1,2$,(i) ${\chi }_i(w)={\chi }_{0,i}>0$ and

${6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}<\frac{\pi \sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}?\frac{2\sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}\arctan ?\frac{d_i?d_3}{2}\sqrt{\frac{n+1}{d_i{d}_3}};$

(ii) $0<{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}\le \frac{d_3}{3(n+1){6{\chi }_{i}6}_{L^{\infty }[0,{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}]}}\min ?\{\frac{2d_i}{d_i+d_3},1\}$.Moreover, we prove asymptotic stabilization of solutions in the sense that:? If $a_1,a_2\in (0,1)$ and $u_0\ne 0\ne v_0$, then any global bounded solution exponentially converge to $(\frac{1?a_1}{1?a_1{a}_2},\frac{1?a_2}{1?a_1{a}_2},0)$ as $t\to \infty$;? If $a_1>1>a_2>0$ and $v_0\ne 0$, then any global bounded solution exponentially converge to $(0,1,0)$ as $t\to \infty$;? If $a_1=1>a_2>0$ and $v_0\ne 0$, then any global bounded solution algebraically converge to $(0,1,0)$ as $t\to \infty$.     

On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.     

A semilinear elliptic equation with a mild singularity at u = 0: Existence and homogenization

In this paper we consider singular semilinear elliptic equations whose prototype is the following

$\{\begin{array}{cc}?divA(x)Du=f(x)g(u)+l(x)\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}?\mathrm{\Omega },\hfill \end{array}$

where Ω is an open bounded set of ${\mathbb{R}}^N,N\ge 1$, $A\in L^{\infty }{(\mathrm{\Omega })}^{N\times N}$ is a coercive matrix, $g:[0,+\infty [\to [0,+\infty ]$ is continuous, and $0\le g(s)\le \frac{1}{s^{\gamma }}+1$ for every $s>0$, with $0<\gamma \le 1$ and $f,l\in L^r(\mathrm{\Omega })$, $r=\frac{2N}{N+2}$ if $N\ge 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, $f(x),l(x)\ge 0$ a.e. $x\in \mathrm{\Omega }$.We prove the existence of at least one nonnegative solution as well as a stability result; we also prove uniqueness if $g(s)$ is nonincreasing or “almost nonincreasing”.Finally, we study the homogenization of these equations posed in a sequence of domains ${\mathrm{\Omega }}^{\epsilon }$ obtained by removing many small holes from a fixed domain Ω.     

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 $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

Local kinetic energy and singularities of the incompressible Navier–Stokes equations

We study the partial regularity problem of the incompressible Navier–Stokes equations. A reverse Hölder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions v belong to $L^{\infty }(0,T;L^{3,w}({\mathbb{R}}^3))$ where $L^{3,w}({\mathbb{R}}^3)$ denotes the standard weak Lebesgue space.     

Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Let $\Omega ?{\mathbb{R}}^N$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ?Ω. We show that the solution to the linear first-order system:(1)$\mathrm{?}\zeta =G\zeta ,\zeta {|}_{\Gamma }=0,$ vanishes if $G\in {\mathsf{L}}^1(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ and $\zeta \in {\mathsf{W}}^{1,1}(\Omega ;{\mathbb{R}}^N)$. In particular, square-integrable solutions ζ of (1) with $G\in {\mathsf{L}}^1\cap {\mathsf{L}}^2(\Omega ;{\mathbb{R}}^{(N\times N)\times N})$ vanish. As a consequence, we prove that:$???:{\mathsf{C}}_{°}^{\infty }(\Omega ,\Gamma ;{\mathbb{R}}^3)\to [0,\infty ),u?6\mathrm{sym}\left(\mathrm{?}u{P}^{?1}\right)6_{{\mathsf{L}}^2(\Omega )}$ is a norm if $P\in {\mathsf{L}}^{\infty }(\Omega ;{\mathbb{R}}^{3\times 3})$ with $\mathrm{Curl}P\in {\mathsf{L}}^p(\Omega ;{\mathbb{R}}^{3\times 3})$, $\mathrm{Curl}{P}^{?1}\in {\mathsf{L}}^q(\Omega ;{\mathbb{R}}^{3\times 3})$ for some $p,q>1$ with $1/p+1/q=1$ as well as $\det P?c^{+}>0$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $\Phi \in {\mathsf{H}}^1(\Omega ;{\mathbb{R}}^3)$, $\Omega ?{\mathbb{R}}^3$, satisfy $\mathrm{sym}(\mathrm{?}{\Phi }^{?}\mathrm{?}\Psi )=0$ for some $\Psi \in {\mathsf{W}}^{1,\infty }(\Omega ;{\mathbb{R}}^3)\cap {\mathsf{H}}^2(\Omega ;{\mathbb{R}}^3)$ with $\det \mathrm{?}\Psi ?c^{+}>0$. Then there exists a constant translation vector $a\in {\mathbb{R}}^3$ and a constant skew-symmetric matrix $A\in \mathfrak{so}(3)$, such that $\Phi =A\Psi +a$.     

Topological bounds for Fourier coefficients and applications to torsion

Let $\mathrm{\Omega }?{\mathbb{R}}^2$ be a bounded convex domain in the plane and consider

$\begin{array}{cc}\hfill ?\mathrm{\Delta }u& =1\text{in}\mathrm{\Omega }\hfill \\ \hfill u& =0\text{on}?\mathrm{\Omega }.\hfill \end{array}$

If u assumes its maximum in $x_0\in \mathrm{\Omega }$, then the eccentricity of level sets close to the maximum is determined by the Hessian $D^{2}u(x_0)$. We prove that $D^{2}u(x_0)$ is negative definite and give a quantitative bound on the spectral gap

$\begin{array}{c}{\lambda }_{\max }(D^{2}u(x_0))\le ?c_1\exp ?(?c_2\frac{\mathrm{diam}(\mathrm{\Omega })}{\mathrm{inrad}(\mathrm{\Omega })})\hfill \\ \text{for universal}{c}_1,c_2>0.\hfill \end{array}$

This is sharp up to constants. The proof is based on a new lower bound for Fourier coefficients whose proof has a topological component: if $f:\mathbb{T}\to \mathbb{R}$ is continuous and has n sign changes, then

$\sum _{k=0}^{n/2}\left|〈f,\sin ?kx〉\right|+\left|〈f,\cos ?kx〉\right|{?}_n\frac{{|f6}_{L^1(\mathbb{T})}^{n+1}}{{6f6}_{L^{\infty }(\mathbb{T})}^n}.$

This statement immediately implies estimates on higher derivatives of harmonic functions u in the unit ball: if u is very flat in the origin, then the boundary function $u(\cos ?t,\sin ?t):\mathbb{T}\to \mathbb{R}$ has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with $f:\mathbb{T}\to \mathbb{R}$ cannot decay faster than $～\exp ?(?{(\mathrm{\#}\text{sign changes})}^{2}t/4)$.     

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.     

Elliptic operators with unbounded diffusion,drift and potential terms

We prove that the realization $A_p$ in $L^p({\mathbb{R}}^N),1<p<\infty$, of the elliptic operator $A=(1+|x{|}^{\alpha })\mathrm{\Delta }+b|x{|}^{\alpha ?1}\frac{x}{|x|}?\mathrm{?}?c|x{|}^{\beta }$ with domain $D(A_p)=\{u\in W^{2,p}({\mathbb{R}}^N)|Au\in L^p({\mathbb{R}}^N)\}$ generates a strongly continuous analytic semigroup $T(?)$ provided that $\alpha >2,\beta >\alpha ?2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [4] and in [16]. Moreover we show that $T(?)$ is consistent, immediately compact and ultracontractive.     

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 $f(x)$.