We present a novel approach for bounding the resolvent of $$H=-\Delta +i(A·\nabla +\nabla ·A)+V=:-\Delta +L1$$ for large energies. It is shown here that there exist a large integer $m$ and a large number ${\lambda }_0$ so that relative to the usual weighted $L^2$-norm, $$\parallel {\left(L{(-\Delta +(\lambda +i0))}^{-1}\right)}^m\parallel <\frac{1}{2}2$$ for all $\lambda >{\lambda }_0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f\left(u\right)=0$ in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\mathrm{𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4\mathrm{𝚘𝚛}N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O\left(2\right)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k\times O(N-2)$ where $D_k\subset O\left(2\right)$ denotes the dihedral...

We consider non-interacting particles subject to a fixed external potential $V$ and a self-generated magnetic field $B$. The total energy includes the field energy $\beta \int B^2$ and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter $\beta$ tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and...

We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_0\in H^{s-1+\epsilon }$, $w_0\in H^{s-1}$ and $b_0\in H^s$ for $s>\frac{3}{2}$ and any $0<\epsilon <1$. The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$.

We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_0$ with values in the Sobolev space $H^s$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i\theta }$ for all $\theta \in ℝ$. We investigate about the persistence of the decorrelation...

We consider the quasilinear equation ${\Delta }_{p}u+K\left(\right|x\left|\right)u^q=0$, and present the proof of the local existence of positive radial solutions near $0$ under suitable conditions on $K$. Moreover, we provide a priori estimates of positive radial solutions near $\infty$ when $r^{-\ell }K\left(r\right)$ for $\ell \ge -p$ is bounded near $\infty$.

We extend a result of the second author [27, Theorem 1.1] to dimensions $d\ge 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<2(d+1)/d-1$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "$ϵ$ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] $L^2$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive...

Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G.$ We prove that the Bruhat-Tits building ${𝔅}^1\left(H\right)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building ${𝔅}^1\left(G\right)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{\prime }$ be maps from ${𝔅}^1\left(H\right)$ to ${𝔅}^1\left(G\right)$ which preserve the Moy–Prasad filtrations....

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3\beta –1}V\left(B^{\beta }\right)$. For the interaction parameter $\beta \in (0,2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N\to \infty$ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0,3/5)$ in [28]. This allows, in the case $\beta \in (0,3/5)$, for the application of the Klainerman–Machedon...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ($P_{\lambda }$) ⎩ $u_{\mid \partial \Omega }=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_{\lambda }$) admits a non-zero, non-negative strong solution $u_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }{\left|\right|}_{W^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto I_{\lambda }\left(u_{\lambda }\right)$ is negative and decreasing in ]0,λ*[, where $I_{\lambda }$ is the energy functional related to ($P_{\lambda }$). ...

Let $ℳf_m$ be the category of $m$-dimensional manifolds and local diffeomorphisms and  let $T$ be the tangent functor on $ℳf_m$. Let $𝒱$ be the category of real vector spaces and linear maps and let ${𝒱}_m$ be the category of $m$-dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F:{𝒱}_m\to 𝒱$ admitting $ℳf_m$-natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$-dimensional manifolds $M$ into almost complex structures $\tilde{J}\left(\nabla \right)$ on $F\left(T\right)M={\bigcup }_{x\in M}F\left(T_{x}M\right)$.

We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy $E_{\epsilon }\left(v\right)=\epsilon {\int }_{\Omega }|{\nabla }^2{v|}^{2}dx+{\epsilon }^{-1}{\int }_{\Omega }{\left(1-\right|\nabla v|}^2{)}^{2}dx$ over $v\in H^2\left(\Omega \right)$, where $\epsilon >0$ is a small parameter. Given $v\in W^{1,\infty }\left(\Omega \right)$ such that $\nabla v\in BV$ and $\left|\nabla v\right|=1$ a.e., we construct a family $\left\{v_{\epsilon }\right\}$ satisfying: $v_{\epsilon }\to v$ in $W^{1,p}\left(\Omega \right)$ and $E_{\epsilon }\left(v_{\epsilon }\right)\to \frac{1}{3}{\int }_{J_{\nabla v}}{\left|{\nabla }^{+}v-{\nabla }^{-}v\right|}^{3}d{\mathscr{H}}^{N-1}$ as $\epsilon$ goes to 0.

A pneumatic tyre in rotating motion with a constant angular velocity $\Omega$ is assimilated to a torus whose generating circle has a radius $R$. The contact of the tyre with the ground is schematized as an ellipse with semi-major axis $a$. When $(\Omega R/C_0)\ll 1$ and $(a/R)\ll 1$ (where $C_0$ is the velocity of the sound), we show that at the rapid time scale $R/C_0$, the air motion within a torus periodically excited on its surface generates an acoustic wave $h$. A study of this acoustic wave is conducted and shows that the mode associated...

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V\left(x\right)\sim {\left|x\right|}^{-\alpha }$, $0<\alpha <2$, and $K\left(x\right)\sim {\left|x\right|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\epsilon }$ belonging to $W^{1,2}\left({ℝ}^N\right)$ is proved under the assumption that $\sigma <p<(N+2)/\left(N-2\right)$ for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that $v_{\epsilon }$ are spikes concentrating at a minimum point of $𝒜=V^{\theta }{K}^{-2/\left(p-1\right)}$, where $\theta =(p+1)/\left(p-1\right)-1/2$.

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

We investigate the recovery of $k$-sparse signals using the ${\ell }_1$-${\ell }_2$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$-sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. $C^p$ solutions in $ℭ$*................................................ 74. Change of variables..............................................................................

We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $(M^n,g)$ be a closed $n$-manifold, $n\ge 3$. The critical Kirchhoff systems we consider are written as $$\left(a+b\sum _{j=1}^p{\int }_M{\left|\nabla u_j\right|}^{2}d{v}_g\right){\Delta }_g{u}_i+\sum _{j=1}^p{A}_{ij}{u}_j={\left|U\right|}^{2^{☆}-2}{u}_i$$ for all $i=1,\cdots ,p$, where ${\Delta }_g$ is the Laplace-Beltrami operator, $A$ is a $C^1$-map from $M$ into the space $M_s^p\left(ℝ\right)$ of symmetric $p\times p$ matrices with real entries, the $A_{ij}$’s are the components of $A$, $U=(u_1,\cdots ,u_p)$, $\left|U\right|:M\to ℝ$ is the Euclidean norm of $U$, $2^{☆}=\frac{2n}{n-2}$ is the critical Sobolev exponent, and...

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma }_T$, $\mathbb{P}\left({\Sigma }_T\right)>0$ such that any data ${\phi }^{\omega }\left(x\right)\in H^{\gamma }\left({𝕋}^3\right),\gamma <1,\omega \in {\Sigma }_T$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-e^{it\Delta }{\phi }^{\omega }\in C([0,T];H^s\left({𝕋}^3\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1\left({𝕋}^3\right)$, that is in the supercritical scaling regime.