Conjugate times and regularity of the minimum time function with differential inclusions

This paper studies the regularity of the minimum time function, $T(\cdot)$, for a control system with a general closed target, taking the state equation in the form of a differential inclusion. Our first result is a sensitivity relation which guarantees the propagation of the proximal subdifferential of $T$ along any optimal trajectory. Then, we obtain the local $C^2$ regularity of the minimum time function along optimal trajectories by using such a relation to exclude the presence of conjugate times

On a system of partial differential equations of Monge-Kantorovich type

We consider a system of PDEs of Monge-Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain $\Omega\subset\mathbb{R}^n$), whose construction is based on an asymmetric Minkowski distance from the boundary of $\Omega$, was already established in [G. Crasta and A. Malusa, The distance function from the boundary in a Minkowski space, to appear in Trans. Amer. Math. Soc.]. In this paper we prove that this solution is essentially unique. A fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form, which is of some interest by itself.Comment: 20 page

Inverse coefficient problem for Grushin-type parabolic operators

The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates, seems hard to apply to the case of Grushin-type operators studied in this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse coefficient problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a strategy that combines Fourier decomposition and Carleman inequalities for certain heat equations with nonsmooth coefficients (solved by the Fourier modes)

Regularity Results for Eikonal-Type Equations with Nonsmooth Coefficients

Solutions of the Hamilton-Jacobi equation $H(x,-Du(x))=1$, with $H(\cdot,p)$ H\"older continuous and $H(x,\cdot)$ convex and positively homogeneous of degree 1, are shown to be locally semiconcave with a power-like modulus. An essential step of the proof is the ${\mathcal C}^{1,\alpha}$-regularity of the extremal trajectories associated with the multifunction generated by $D_pH$

Generalized characteristics and Lax-Oleinik operators: global theory

For autonomous Tonelli systems on $\R^n$, we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics

Well-posedness for a class of nonlinear degenerate parabolic equations

In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces

Exterior sphere condition and time optimal control for differential inclusions

The minimum time function $T(\cdot)$ of smooth control systems is known to be locally semiconcave provided Petrov's controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of $T(\cdot)$ near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened set-up, we prove that the hypograph of $T(\cdot)$ satisfies, locally, an exterior sphere condition. As is well-known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming $T(\cdot)$ to be Lipschitz

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