Regularity results for the minimum time function with H\"ormander vector fields

In a bounded domain of $\mathbb{R}^n$ with smooth boundary, we study the regularity of the viscosity solution, $T$, of the Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\ldots ,X_N\}$, subject to H\"ormander's bracket generating condition. Due to the presence of characteristic boundary points, singular trajectories may occur in this case. We characterize such trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. We then prove that the local Lipschitz continuity of $T$, the local semiconcavity of $T$, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied when the characteristic set of $\{X_1,\ldots ,X_N\}$ is a symplectic manifold. We apply our results to Heisenberg's and Martinet's vector fields

Some properties of semiconcave functions with general modulus

AbstractThe class of semiconcave functions represents a useful generalization of the one of concave functions. Such an extension can be achieved requiring that a function satisfies a suitable one-sided estimate. In this paper, the structure of the set of points at which a semiconcave function fails to be differentiable—the singular set—is studied. First, we prove some results on the existence of arcs contained on the singular set. Then, we show how these abstract results apply to semiconcave solutions of Hamilton–Jacobi equations

On the subelliptic eikonal equation

On a bounded smooth domain, we consider the viscosity solution of the homogeneous Dirichlet problem for the eikonal equation associated with a system of Hoermander\u2019s vector fields. We present some results on the regularity and the structure of the singular support of such a function

Analytic Hypoellipticity in the Presence of Lower Order Terms

We consider a second order operator with analytic coefficients whose principal symbol vanishes exactly to order two on a symplectic real analytic manifold. We assume that the first (non degenerate) eigenvalue vanishes on a symplectic submanifold of the characteristic manifold. In the $C^\infty$ framework this situation would mean a loss of 3/2 derivatives. We prove that this operator is analytic hypoelliptic. The main tool is the FBI transform. A case in which $C^\infty$ hypoellipticity fails is also discussed.Comment: 40 page

Bad neighbors? Niche overlap and asymmetric competition between native and Lessepsian limpets in the Eastern Mediterranean rocky intertidal.

Abstract The Eastern Mediterranean Sea hosts more non-indigenous species than any other marine region, yet their impacts on the native biota remain poorly understood. Focusing on mollusks from the Israeli rocky intertidal, we explored the hypothesis that this abiotically harsh habitat supports a limited trait diversity, and thus may promote niche overlap and competition between native and non-indigenous species. Indeed, native and non-indigenous assemblage components often had a highly similar trait composition, caused by functionally similar native (Patella caerulea) and non-indigenous (Cellana rota) limpets. Body size of P. caerulea decreased with increasing C. rota prevalence, but not vice versa, indicating potential asymmetric competition. Although both species have coexisted in Israel for >15 years, a rapid 'replacement' of native limpets by C. rota has been reported for a thermally polluted site, suggesting that competition and regionally rapid climate-related seawater warming might interact to progressively erode native limpet performance along the Israeli coast

Comunidades de moluscos de los estratos foliares y rizómicos de un asentamiento profundo de Posidonia oceanica en el Tirreno central

The molluscan assemblages inhabiting the leaf and rhizome layers of Posidonia oceanica were studied in a deep water (–24/26 m) settlement of a highly heterogeneous substratum on an off-shore reef in the central Tyrrhenian Sea. This is one of the few works dealing with the rhizome layer and with Posidonia oceanica that has settled on hard substrata. The leaf assemblage only had a few species, less than other assemblages at a comparable depth in the same basin. This poorness may be due to the depth, but it may also be due to the high fragmentation of the meadow. However, the high percentage of carnivores is consistent with previous observations in deep water meadows. The rhizome assemblage is highly diverse both in terms of species and feeding guilds, which could be explained by the higher affinity for low light conditions of most molluscs and the greater habitat heterogeneity. The marked differences in the two taxocoenoses and the high diversification of the rhizome assemblage evidence that they should be included in studies on the potential diversity of Posidonia oceanica meadows. Failure to consider this layer seriously affects any evaluation of the biodiversity of this habitat, which is of great conservation interest.Las comunidades de moluscos de los estratos foliares y rizómicos de Posidonia oceanica fueron estudiadas en un asentamiento sobre un sustrato altamente heterogéneo de un arrecife costero del mar Tirreno central y de aguas profundas (–24/26 m). Este es uno de los escasos trabajos publicados hasta el momento del asentamiento de Posidonia oceanica sobre sustratos duros y de su correspondiente estrato rizómico. La agrupación de hojas tiene solo unas pocas especies, incluso menos que otras agrupaciones de una profundidad comparable en la misma cuenca. Esta pobreza podría estar causada por la profundidad pero también por la gran fragmentación de la pradera. Por el contrario, el alto porcentaje de carnívoros es consistente con observaciones previas en praderas de aguas profundas. La agrupación de los rizomas es altamente diversa, tanto en términos de especies como de categoría trófica y esto podría ser explicado por la mayor afinidad de la mayoría de los moluscos hacia condiciones de baja luminosidad y la gran heterogeneidad de hábitat. Las marcadas diferencias en las dos taxocenosis y la gran diversificación de las especies del grupo de rizomas aumenta la necesidad de ser incluidos en estudios sobre diversidad potencial de las praderas de Posidonia oceanica. No considerar este estrato afectará seriamente cualquier evaluación de la biodiversidad de este hábitat de gran interés para la conservación