Harnack's inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack's inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach from [DGL08] to obtain Harnack's inequality

Maxwell equations in distributional sense and applications

This paper discusses the Maxwell system of electrodynamics in the context of distributions. It is used to establish boundary conditions for fields at the interface when the charge and current densities are concentrated measures on the interface. The paper derives the generalized Snell's law from this analysis.Comment: 15 page

Sobolev inequalities with jointly concave weights on convex cones

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form $$\left(\int_E |u(x)|^q\,\omega(x) \,dx\right)^{1/q}\leq K_0\,\left(\int_E |\nabla u(x)|^p\,\sigma(x)\,dx\right)^{1/p},\ \ u\in C_0^\infty(\mathbb R^n),\ \ \ \ \ \ {\rm (WSI)}$$ where $p\geq 1$ and $q>0$ is the corresponding Sobolev critical exponent. Here $E\subseteq \mathbb R^n$ is an open convex cone, and $\omega,\sigma:E\to (0,\infty)$ are two homogeneous weights verifying a general concavity-type structural condition. The constant $K_0= K_0(n, p, q, \omega, \sigma) >0$ is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that $K_0$ is optimal in (WSI) if and only if $\omega$ and $\sigma$ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to PDEs are also provided.Comment: 35 pages; some references are updated. To appear in the Proceedings of the London Mathematical Societ

Metasurfaces and Optimal transport

This paper provides a theoretical and numerical approach to show existence, uniqueness, and the numerical determination of metalenses refracting radiation with energy patterns. The theoretical part uses ideas from optimal transport and for the numerical solution we study and implement a damped Newton algorithm to solve the semi discrete problem. A detailed analysis is carried out to solve the near field one source refraction problem and extensions to the far field are also mentioned.Comment: 29 pages, 4 figure

Estudio comparativo de las marcas de dientes producidas por dos pequeños carnívoros sudamericanos

In the following paper we present the preliminary results of an experimental study performed with Pampas fox (Lycalopex gymnocercus) and Geoffroy’s cat (Leopardus geoffroyi). The objectives are to characterize the tooth mark patterns generated by each carnivore on non ingested bone of a small mammal and evaluate if there exists differences in these patterns. Results indicate that both carnivores generate similar types and proportions of modifications; however, the average number of marks per specimen is double for the Pampas fox. The preliminary information obtained here indicates that the size of the pits does not appear to be a sufficient diagnostic criterion to distinguish the action of these two predators.En este trabajo se presentan los primeros resultados de un estudio experimental realizado con zorro pampeano (Lycalopex gymnocercus) y gato montés (Leopardus geoffroyi) con el fin de caracterizar el patrón de marcas de dientes generado por cada carnívoro sobre restos óseos no ingeridos de un mamífero pequeño y evaluar si existen diferencias en estos patrones. Este estudio indicó que ambos carnívoros generan los mismos tipos de modificaciones y en proporciones similares; no obstante, el número promedio de marcas por espécimen es más del doble en el caso del zorro. Sobre la base de los resultados preliminares obtenidos se propone que el tamaño de los hoyuelos no parece ser un criterio diagnóstico determinante para distinguir la acción de estos dos predadores