Existence of solutions of a non-linear eigenvalue problem with a variable weight

We study the non-linear minimization problem on $H_0^1(\mathrm{\Omega })?L^q$ with $q=\frac{2n}{n?2}$, $\alpha >0$ and $n\ge 4$:

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }a(x,u)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2$

where $a(x,s)$ presents a global minimum α at $(x_0,0)$ with $x_0\in \mathrm{\Omega }$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behavior of $a(x,s)$ with respect to s. The model case is

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }(\alpha +|x{|}^{\beta }|u{|}^k)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2.$

In a previous paper dedicated to the same problem with $\lambda =0$, we showed that minimizers exist only in the range $\beta <kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta \ge kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\mathrm{\Omega })$, $0\le k\le q?2$ and $\beta >kn/q+2$, minimizers do exist.     

On the representation as exterior differentials of closed forms with L1-coefficients

Let $N\ge 2$. If $g\in L_c^1({\mathbf{R}}^N)$ has zero integral, then the equation $\mathrm{div}X=g$ need not have a solution $X\in W_{\mathrm{loc}}^{1,1}({\mathbf{R}}^N;{\mathbf{R}}^N)$ [6] or even $X\in L_{\mathrm{loc}}^{N/(N?1)}$ $({\mathbf{R}}^N;{\mathbf{R}}^N)$ [2]. Using these results, we prove that, whenever $N\ge 3$ and $2\le ?\le N?1$, there exists some ?-form $f\in L_c^1({\mathbf{R}}^N;{\mathrm{\Lambda }}^{?})$ such that $\mathrm{d}f=0$ and the equation $\mathrm{d}\lambda =f$ has no solution $\lambda \in W_{\mathrm{loc}}^{1,1}({\mathbf{R}}^N;{\mathrm{\Lambda }}^{??1})$. This provides a negative answer to a question raised by Baldi, Franchi, and Pansu [1].     

SL(m,C)-equivariant and translation covariant continuous tensor valuations

The space of continuous, $\mathrm{SL}(m,\mathbb{C})$-equivariant, $m\ge 2$, and translation covariant valuations taking values in the space of real symmetric tensors on ${\mathbb{C}}^m?{\mathbb{R}}^{2m}$ of rank $r\ge 0$ is completely described. The classification involves the moment tensor valuation for $r\ge 1$ and is analogous to the known classification of the corresponding tensor valuations that are $\mathrm{SL}(2m,\mathbb{R})$-equivariant, although the method of proof cannot be adapted.     

Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion

This paper studies the asymptotic behavior of smooth solutions to the generalized Hall-magneto-hydrodynamics system (1.1) with one single diffusion on the whole space ${\mathbb{R}}^3$. We establish that, in the inviscid resistive case, the energy ${6b(t)6}_2^2$ vanishes and ${6u(t)6}_2^2$ converges to a constant as time tends to infinity provided the velocity is bounded in $W^{1?\alpha ,\frac{3}{\alpha }}({\mathbb{R}}^3)$; in the viscous non-resistive case, the energy ${6u(t)6}_2^2$ vanishes and ${6b(t)6}_2^2$ converges to a constant provided the magnetic field is bounded in $W^{1?\beta ,\infty }({\mathbb{R}}^3)$. In summary, one single diffusion, being as weak as ${(?\mathrm{\Delta })}^{\alpha }b$ or ${(?\mathrm{\Delta })}^{\beta }u$ with small enough $\alpha ,\beta$, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.     

Classification of metabelian 2-groups G with Gab ≃ (2,2n),n ≥ 2, and rank d(G′)=2; Applications to real quadratic number fields

We characterize all finite metabelian 2-groups G whose abelianizations $G^{\mathrm{ab}}$ are of type $(2,2^n)$, with $n\ge 2$, and for which their commutator subgroups $G^{\prime }$ have $\text{rank}=2$. This is given in terms of the order of the abelianizations of the maximal subgroups and the structure of the abelianizations of those normal subgroups of index 4 in G. We then translate these group theoretic properties to give a characterization of number fields k with 2-class group ${\mathrm{Cl}}_2(k)?(2,2^n)$, $n\ge 2$, such that the rank of ${\mathrm{Cl}}_2(k^1)=2$ where $k^1$ is the Hilbert 2-class field of k. In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.     

Embedding of RCD⁎(K,N) spaces in L2 via eigenfunctions

In this paper we study the family of embeddings ${\mathrm{\Phi }}_t$ of a compact ${\mathrm{RCD}}^{?}(K,N)$ space $(X,\mathsf{d},\mathfrak{m})$ into $L^2(X,\mathfrak{m})$ via eigenmaps. Extending part of the classical results [10], [11] known for closed Riemannian manifolds, we prove convergence as $t\downarrow 0$ of the rescaled pull-back metrics ${\mathrm{\Phi }}_t^{?}{g}_{L^2}$ in $L^2(X,\mathfrak{m})$ induced by ${\mathrm{\Phi }}_t$. Moreover we discuss the behavior of ${\mathrm{\Phi }}_t^{?}{g}_{L^2}$ with respect to measured Gromov-Hausdorff convergence and t. Applications include the quantitative $L^p$-convergence in the noncollapsed setting for all $p<\infty$, a result new even for closed Riemannian manifolds and Alexandrov spaces.