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

Partial Lipschitz regularity of the minimum time function for sub-Riemannian control systems

In Euclidean space of dimension 2 or 3, we study a minimum time problem associated with a system of real-analytic vector fields satisfying H\"ormander's bracket generating condition, where the target is a nonempty closed set. We show that, in dimension 2, the minimum time function is locally Lipschitz continuous while, in dimension 3, it is Lipschitz continuous in the complement of a set of measure zero. In particular, in both cases, the minimum time function is a.e. differentiable on the complement of the target. In dimension 3, in general, there is no hope to have the same regularity result as in dimension 2. Indeed, examples are known where the minimum time function fails to be locally Lipschitz continuous.Comment: 18 pages, no figure

Existence and asymptotic behavior for L^2-norm preserving nonlinear heat equations

We consider a nonlinear parabolic equation with a nonlocal term, which preserves the L^2-norm of the solution. We study the local and global well posedness on a bounded domain, as well as the whole Euclidean space, in H^1. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H^1 to a stationary state. For a ball, we prove strong asymptotic convergence to the ground state when the initial condition is positive

Challenging management of gingival squamous cell carcinoma:a 10 years single-center retrospective study on Northern-Italian patients

Aim of this study was to describe the outcome of patients with gingival squamous cell carcinoma (GSCC), and to recognize aspects affecting clinical course and to consider survival rate. The case records of patients, over a 10-year period, were retrospectively examined. Differences in distribution of the potential risk factors by prognosis were investigated through non-parametrical tests (Wilcoxon Rank-Sum and Fisher?s Exact). Survival curves for age, therapy and stage were built by the Kaplan-Meier method and compared with Log-Rank test. 79 patients were analysed. Significant increase in mortality for patients older than 77 and for those with advanced stages was found. Cumulative survival rate 5 years after the diagnosis was 43%, while at 10 years was of 11%. With a statistical relationship between age and tumour stage with survival rates, and 70% of GSCC cases identified as stage IV, early GSCC diagnosis remains challenging

CMS physics technical design report : Addendum on high density QCD with heavy ions

Peer reviewe