Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite measures which do not charge sets of sigma-finite Hausdorff measure H^{N-1} can be written as an L^1 perturbation of the divergence of a continuous vector field. The main tool is a property of approximation of measures in terms of the Hausdorff content

Flat solutions of the 1-Laplacian equation

For every $f \in L^N(\Omega)$ defined in an open bounded subset $\Omega$ of $\mathbb{R}^N$, we prove that a solution $u \in W_0^{1, 1}(\Omega)$ of the $1$-Laplacian equation ${-}\mathrm{div}{(\frac{\nabla u}{|\nabla u|})} = f$ in $\Omega$ satisfies $\nabla u = 0$ on a set of positive Lebesgue measure. The same property holds if $f \not\in L^N(\Omega)$ has small norm in the Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.Comment: Dedicated to Jean Mawhin. Revised and extended version of a note written by the authors in 201

Strong maximum principle for Schr\"odinger operators with singular potential

We prove that for every $p > 1$ and for every potential $V \in L^p$, any nonnegative function satisfying $-\Delta u + V u \ge 0$ in an open connected set of $\mathbb{R}^N$ is either identically zero or its level set $\{u = 0\}$ has zero $W^{2, p}$ capacity. This gives an affirmative answer to an open problem of B\'enilan and Brezis concerning a bridge between Serrin-Stampacchia's strong maximum principle for $p > \frac{N}{2}$ and Ancona's strong maximum principle for $p = 1$. The proof is based on the construction of suitable test functions depending on the level set $\{u = 0\}$ and on the existence of solutions of the Dirichlet problem for the Schr\"odinger operator with diffuse measure data.Comment: 21 page

Schroedinger operators involving singular potentials and measure data

We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data $$\left\{ \begin{alignedat}{2} -\Delta u + Vu & = \mu && \quad \text{in } \Omega,\\ u & = 0 && \quad \text{on } \partial \Omega. \end{alignedat} \right.$$ We characterize the finite measures $\mu$ for which this problem has a solution for every nonnegative potential $V$ in the Lebesgue space $L^p(\Omega)$ with $1 \le p \le N/2$. The full answer can be expressed in terms of the $W^{2,p}$ capacity for $p > 1$, and the $W^{1,2}$ (or Newtonian) capacity for $p = 1$. We then prove the existence of a solution of the problem above when $V$ belongs to the real Hardy space $H^1(\Omega)$ and $\mu$ is diffuse with respect to the $W^{2,1}$ capacity.Comment: Fixed a display problem in arxiv's abstract. Original tex file unchange

A note on the fractional perimeter and interpolation

We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces $W^{\alpha, 1}$ of order $0 < \alpha < 1$

Limit solutions of the Chern-Simons equation

We investigate the scalar Chern-Simons equation $-\Delta u + e^u(e^u-1) = \mu$ in cases where there is no solution for a given nonnegative finite measure $\mu$. Approximating $\mu$ by a sequence of nonnegative $L^1$ functions or finite measures for which this equation has a solution, we show that the sequence of solutions of the Dirichlet problem converges to the solution with largest possible datum \mu^# \le \mu and we derive an explicit formula of \mu^# in terms of $\mu$. The counterpart for the Chern-Simons system with datum $(\mu, \nu)$ behaves differently and the conclusion depends on how much the measures $\mu$ and $\nu$ charge singletons

Reactive oxygen species, vascular noxs, and hypertension: focus on translational and clinical research

Significance: Reactive oxygen species (ROS) are signaling molecules that are important in physiological processes, including host defense, aging, and cellular homeostasis. Increased ROS bioavailability and altered redox signaling (oxidative stress) have been implicated in the onset and/or progression of chronic diseases, including hypertension. Recent Advances: Although oxidative stress may not be the only cause of hypertension, it amplifies blood pressure elevation in the presence of other pro-hypertensive factors, such as salt loading, activation of the renin-angiotensin-aldosterone system, and sympathetic hyperactivity, at least in experimental models. A major source for ROS in the cardiovascular-renal system is a family of nicotinamide adenine dinucleotide phosphate oxidases (Noxs), including the prototypic Nox2-based Nox, and Nox family members: Nox1, Nox4, and Nox5. Critical Issues: Although extensive experimental data support a role for increased ROS levels and altered redox signaling in the pathogenesis of hypertension, the role in clinical hypertension is unclear, as a direct causative role of ROS in blood pressure elevation has yet to be demonstrated in humans. Nevertheless, what is becoming increasingly evident is that abnormal ROS regulation and aberrant signaling through redox-sensitive pathways are important in the pathophysiological processes which is associated with vascular injury and target-organ damage in hypertension. Future Directions: There is a paucity of clinical information related to the mechanisms of oxidative stress and blood pressure elevation, and a few assays accurately measure ROS directly in patients. Such further ROS research is needed in humans and in the development of adequately validated analytical methods to accurately assess oxidative stress in the clinic

The role of interplay between coefficients in the GG-convergence of some elliptic equations

We study the behavior of the solutions $u$ of the linear Dirichlet problems $- \mathrm{div} (M(x) \nabla u) + a(x) u = f(x)$ with respect to perturbations of the matrix $M(x)$ (with respect to the $G$-convergence) and with respect to perturbations of the nonnegative coefficient $a(x)$ and of the right hand side $f(x)$ satisfying the condition $|f (x)| \leq Q \, a (x)$