This paper gives a new proof of the fact that a $k$-dimensional normal current $T$ in ${ℝ}^m$ is integer multiplicity rectifiable if and only if for every projection $P$ onto a $k$-dimensional subspace, almost every slice of $T$ by $P$ is $0$-dimensional integer multiplicity rectifiable, in other words, a sum of Dirac masses with integer weights. This is a special case of the Rectifiable Slices Theorem, which was first proved a few years ago by B. White.

Two new time-dependent versions of div-curl results in a bounded domain $\Omega \subset {ℝ}^3$ are presented. We study a limit of the product $v_k{w}_k$, where the sequences $v_k$ and $w_k$ belong to ${Ł}_2\left(\Omega \right)$. In Theorem 2.1 we assume that $\nabla \times v_k$ is bounded in the $L_p$-norm and $\nabla ·w_k$ is controlled in the $L_r$-norm. In Theorem 2.2 we suppose that $\nabla \times w_k$ is bounded in the $L_p$-norm and $\nabla ·w_k$ is controlled in the $L_r$-norm. The time derivative of $w_k$ is bounded in both cases in the norm of $-1\left(\Omega \right)$. The convergence (in the sense of distributions) of $v_k{w}_k$ to the product $vw$...

Let $\Omega$ be a smooth bounded domain in ${ℝ}^N,N\>1$ and let $n\in {\mathbb{N}}^{*}$. We prove here the existence of nonnegative solutions $u_n$ in $BV\left(\Omega \right)$, to the problem $$\left(P_n\right)\left\{\begin{array}{cc}-div\sigma +2n\left({\int }_{\Omega }u-1\right){sign}^{+}\left(u\right)=0\hfill & \text{in}\Omega ,\hfill \\ \sigma ·\nabla u=\left|\nabla u\right|\hfill & \text{in}\Omega ,\hfill \\ u\text{is}\text{not}\text{identically}\text{zero},-\sigma ·\overrightarrow{n}u=u\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\overrightarrow{n}$ denotes the unit outer normal to $\partial \Omega$, and ${\mathrm{sign}}^{+}\left(u\right)$ denotes some $L^{\infty }\left(\Omega \right)$ function defined as: $${\mathrm{sign}}^{+}\left(u\right).u=u^{+},0\le {\mathrm{sign}}^{+}\left(u\right)\le 1.$$ Moreover, we prove the tight convergence of $u_n$ towards one of the first eingenfunctions for the first $1-$Laplacian Operator $-{\Delta }_1$ on $\Omega$ when $n$ goes to $+\infty$.

Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the $p^2$-rank of the class group as a quantity of relevance in the $p$-class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class...

We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here $f\in L^1\left(\Omega \right)$ and the solution belongs to the so-called grand Sobolev space $W_0^{1,2)}\left(\Omega \right)$. This is the proper space when the right hand side is assumed to be only $L^1$-integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li.

In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than $L^2$, extending the $L^2$ well-posedness result of Molinet [20].

We consider non-linear elliptic equations having a measure in the right-hand side, of the type $diva(x,Du)=\mu ,$ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calderón-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori...

Let $M$ be a differentiable manifold with a pseudo-Riemannian metric $g$ and a linear symmetric connection $K$. We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on $TM$ generated by $g$ and $K$. We get that all natural vector fields are of the form $$E\left(u\right)=\alpha \left(h\left(u\right)\right)u^H+\beta \left(h\left(u\right)\right)u^V,$$ where $u^V$ is the vertical lift of $u\in T_{x}M$, $u^H$ is the horizontal lift of $u$ with respect to $K$, $h\left(u\right)=1/2g(u,u)$ and $\alpha ,\beta$ are smooth real functions defined on $R$. All natural 2-vector fields are of the form $$\Lambda \left(u\right)={\gamma }_1\left(h\left(u\right)\right)\Lambda (g,K)+{\gamma }_2\left(h\left(u\right)\right)u^H\wedge u^V,$$ where ${\gamma }_1$, ${\gamma }_2$ are smooth real functions...