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

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 $0 < \alpha < 1$

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

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

Performance of a New Enhanced Topological Decision-Rule Map-Matching Algorithm for Transportation Applications

Indexación: Web of Science; ScieloMap-matching problems arise in numerous transportation-related applications when spatial data is collected using inaccurate GPS technology and integrated with a flawed digital roadway map in a GIS environment. This paper presents a new enhanced post-processing topological decision-rule map-matching algorithm in order to address relevant special cases that occur in the spatial mismatch resolution. The proposed map-matching algorithm includes simple algorithmic improvements: dynamic buffer that varies its size to snap GPS data points to at least one roadway centerline; a comparison between vehicle heading measurements and associated roadway centerline direction; and a new design of the sequence of steps in the algorithm architecture. The original and new versions of the algorithm were tested on different spatial data qualities collected in Canada and United States. Although both versions satisfactorily resolve complex spatial ambiguities, the comparative and statistical analysis indicates that the new algorithm with the simple algorithmic improvements outperformed the original version of the map-matching algorithm.El problema de la ambigüedad espacial ocurre en varias aplicaciones relacionadas con transporte, específicamente cuando existe inexactitud en los datos espaciales capturados con tecnología GPS o cuando son integrados con un mapa digital que posee errores en un ambiente SIG. Este artículo presenta un algoritmo nuevo y mejorado basado en reglas de decisión que es capaz de resolver casos especiales relevantes en modo post-proceso. El algoritmo propuesto incluye las siguientes mejoras algorítmicas: un área de búsqueda dinámica que varía su tamaño para asociar puntos GPS a al menos un eje de calzada, una comparación entre el rumbo del vehículo y la dirección del eje de calzada asignada, y un nuevo diseño de la secuencia de pasos del algoritmo. Tanto el algoritmo original como el propuesto fueron examinados con datos espaciales de diferentes calidades capturados en Canadá y Estados Unidos. Aunque ambas versiones resuelven satisfactoriamente el problema de ambigüedad espacial, el análisis comparativo y estadístico indica que la nueva versión del algoritmo con las mejoras algorítmicas entrega resultados superiores a la versión original del algoritmo.http://ref.scielo.org/9mt55