153 research outputs found
Regularity Results for Eikonal-Type Equations with Nonsmooth Coefficients
Solutions of the Hamilton-Jacobi equation , with
H\"older continuous and 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 -regularity of the
extremal trajectories associated with the multifunction generated by
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)
Generalized characteristics and Lax-Oleinik operators: global theory
For autonomous Tonelli systems on , 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
Exterior sphere condition and time optimal control for differential inclusions
The minimum time function 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 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
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 to be Lipschitz
Regularity results for the minimum time function with H\"ormander vector fields
In a bounded domain of with smooth boundary, we study the
regularity of the viscosity solution, , of the Dirichlet problem for the
eikonal equation associated with a family of smooth vector fields , 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 , the local semiconcavity of , and the
absence of singular trajectories are equivalent properties. Finally, we show
that the last condition is satisfied when the characteristic set of
is a symplectic manifold. We apply our results to
Heisenberg's and Martinet's vector fields
Global Propagation of Singularities for Time Dependent Hamilton-Jacobi Equations
We investigate the properties of the set of singularities of semiconcave
solutions of Hamilton-Jacobi equations of the form \begin{equation*}
u_t(t,x)+H(\nabla u(t,x))=0, \qquad\text{a.e. }(t,x)\in
(0,+\infty)\times\Omega\subset\mathbb{R}^{n+1}\,. \end{equation*} It is well
known that the singularities of such solutions propagate locally along
generalized characteristics. Special generalized characteristics, satisfying an
energy condition, can be constructed, under some assumptions on the structure
of the Hamiltonian . In this paper, we provide estimates of the dissipative
behavior of the energy along such curves. As an application, we prove that the
singularities of any viscosity solution of the above equation cannot vanish in
a finite time
Propagation of singularities for weak KAM solutions and barrier functions
This paper studies the structure of the singular set (points of
nondifferentiability) of viscosity solutions to Hamilton-Jacobi equations
associated with general mechanical systems on the n-torus. First, using the
level set method, we characterize the propagation of singularities along
generalized characteristics. Then, we obtain a local propagation result for
singularities of weak KAM solutions in the supercritical case. Finally, we
apply such a result to study the propagation of singularities for barrier
functions
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