We define various classes of Sobolev bundles and connections and study their topological and analytical properties. We show that certain kinds of topologies (which depend on the classes) are well-defined for such bundles and they are stable with respect to the natural Sobolev topologies. We also extend the classical Chern-Weil theory for such classes of bundles and connections. Applications related to variational problems for the Yang-Mills functional are also given.

Given an immersed submanifold $M^n\subset\mathbb{R}^{n+2}$, we characterize the vanishing of the normal curvature $R_D$ at a point $p \in M$ in terms of the behaviour of the asymptotic directions and the curvature locus at $p$. We relate the affine properties of codimension 2 submanifolds with flat normal bundle with the conformal properties of hypersurfaces in Euclidean space. We also characterize the semiumbilical, hypespherical and conformally flat submanifolds of codimension 2 in terms of their curvature loci.

Let ${\mathcal X}^\dagger$ be a smooth $\dagger$-scheme (in the sense of Meredith) over a complete discrete valuation ring $(V, {\mathfrak m})$ of unequal characteristics $(0,p)$ and let ${\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}$ be the sheaf of $V$-linear endomorphisms of ${\mathcal O}_{{\mathcal X}^\dagger}$ whose reduction modulo ${\mathfrak m}^s$ is a linear differential operator of order bounded by an affine function in $s$. In this paper we prove that locally there is an ${\mathcal O}_{{\mathcal X}^\dagger}$-isomorphism between the sections of ${\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}$ and the overconvergent total symbols, and we deduce a cohomological triviality property.

Let $(X,\mathcal{B}, \mu, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $$ R^* : (f, g) \in L^1 \times L^1 \rightarrow R^*(f, g)(x) = \sup_{n} \frac{f(T^n x) g(T^{2n} x)}{n}. $$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy-Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1$. The Furstenberg averages do not converge for all pairs of $(L^1,L^1)$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^p,L^q)$ functions with $\frac{1}{p}+\frac{1}{q} \leq 1$.

We consider the Keller-Segel system of degenerate type (KS)$_m$ with $m > 1$ below. We establish a uniform estimate of $\partial_x^2 u^{m-1}$ from below. The corresponding estimate to the porous medium equation is well-known as an Aronson-Bénilan type. We apply our estimate to prove the optimal Hölder continuity of weak solutions of (KS)$_m$. In addition, we find that the set $D(t):=\{ x \in \mathbb{R}; u(x,t) > 0\}$ of positive region to the solution $u$ is monotonically non-decreasing with respect to $t$.

In this paper, we prove the local well-posedness in critical Besov spaces for the compressible Navier-Stokes equations with density dependent viscosities under the assumption that the initial density is bounded away from zero.

We extend an $L^2$ maximal multiplier result of Bourgain to all $L^p$ spaces, $1 < p < \infty$.

We study the overdetermined problem $$ \left\{ \begin{array}{cc} \Delta u + f(u) = 0 & \mbox{ in $\Omega$,} \\ u = 0 & \mbox{ on $\partial\Omega$,} \\ \partial_\nu u = c & \mbox{ on $\Gamma$,} \end{array} \right. $$ where $\Omega$ is a locally Lipschitz epigraph, that is $C^3$ on $\Gamma\subseteq\partial\Omega$, with $\partial\Omega\setminus\Gamma$ consisting in nonaccumulating, countably many points. We provide a geometric inequality that allows us to deduce geometric properties of the sets $\Omega$ for which monotone solutions exist. In particular, if $\mathcal{C} \in \mathbb{R}^n$ is a cone and either $n=2$ or $n=3$ and $f \ge 0$, then there exists no solution of $$ \left\{ \begin{array}{cc} \Delta u + f(u) = 0 & \mbox{ in $\mathcal{C}$,} \\ u > 0 & \mbox{ in $\mathcal{C}$,} \\ u = 0 & \mbox{ on $\partial\mathcal{C}$,} \\ \partial_\nu u = c & \mbox{ on $\partial\mathcal{C} \setminus \{0\}$.} \end{array} \right. $$ This answers a question raised by Juan Luis Vázquez.

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.] of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.

In this paper we settle (in dimension $n=2$) the open question whether for a divergence form equation $\div (A\nabla u) = 0$ with coefficients satisfying certain minimal smoothness assumption (a Carleson measure condition), the $L^p$ Neumann and Dirichlet regularity problems are solvable for some values of $p\in (1,\infty)$. The related question for the $L^p$ Dirichlet problem was settled (in any dimension) in 2001 by Kenig and Pipher [Kenig, C.E. and Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45 (2001), no. 1, 199-217].

We are interested in characterizing the compact sets of the Heisenberg group that are contained in a curve of finite length. Ferrari, Franchi and Pajot recently gave a sufficient condition for those sets, adapting a necessary and sufficient condition due to P. Jones in the Euclidean setting. We prove that this condition is not necessary.

Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $\varphi: M \rightarrow \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower bound for the width of the cone in terms of the energy and the tension of $\varphi$ and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case $\varphi$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $\varphi(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$.

We prove a variant of the classical Whitney extension theorem, in which the $C^m$-norm of the extending function is controlled up to a given, small percentage error.