Abstracts of preprints
Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of space dependent Aronsson Equations, Pierpaolo Soravia
{abstract}
In this paper we study absolute minimizers and the Aronsson equation for a noncoercive Hamiltonian. We extend the definition of absolutely minimizing functions (in a viscosity sense) for the minimization of the $L^\infty$ norm of a Hamiltonian, within a class of locally Lipschitz continuous functions with respect to possibly noneuclidian metrics. The metric structure is naturally associated to the Hamiltonian and it is related to the a-priori regularity of the family of subsolutions of the Hamilton-Jacobi equation. A special but relevant case contained in our framework is that of Hamiltonians with a Carnot-Carat\`eodory metric structure determined by a family of vector fields, in particular the eikonal Hamiltonian and the corresponding anisotropic infinity-Laplace equation. In this case, the definition of absolute minimizer can be written in an almost classical way, by the theory of Sobolev spaces in a CC setting. In general open domains and with a prescribed continuous Dirichlet boundary condition, we prove the existence of an absolute minimizer and derive the Aronsson equation as a viscosity solution for such minimizer. The proof is based on Perron's method and relies on a-priori continuity estimates for absolute minimizers.
Viscosity and almost everywhere solutions of first-order Carnot-Carath\`eodory Hamilton-Jacobi equations, Pierpaolo Soravia
{abstract} We consider viscosity and distributional derivatives of functions in the directions of a family of vector fields, generators of a Carnot-Carath\`eodory metric. In the framework of convex and non coercive Hamilton-Jacobi equations of Carnot-Carath\`eodory type (C-C in brief) we show that viscosity and a.e. subsolutions are equivalent concepts. The latter is a concept related to Lipschitz continuity with respect to the metric generated by the family of vector fields. Under more restrictive assumptions that include Carnot groups, we prove that viscosity solutions of C-C HJ equations are Lipschitz continuous with respect to the corresponding Carnot-Carath\`eodory metric
and satisfy the equation a.e..
Cauchy problems for noncoercive Hamilton-Jacobi equations with discontinuous coefficients, Cecilia De Zan and Pierpaolo Soravia
{abstract} We prove that under suitable assumptions the Cauchy problem for an homogeneous, but not necessarily coercive, Hamilton-Jacobi-Isaacs equation with an $x$-dependent, piecewise continuous coefficient, admits a unique and continuous viscosity solution. The result applies in particular to the Carnot-Carat\`eodory eikonal equation with discontinuous refraction index of a family of vector fields satisfying the H\"ormander condition. Our results have also interest to define geometric flows with discontinuous velocity in anisotropic media with a non-euclidian ambient space.
Generalized Hessians of $C^{1,1}$-functions and second-order viscosity subjets,
Luc BARBET, Aris DANIILIDIS, Pierpaolo SORAVIA
{Abstract.} Given a $C^{1,1}$--function $f:U\rightarrow
\mathbb{R}$ (where $U\subset\mathbb{R}^{n}$ open) we deal with the question of
whether or not at a given $x_{0}\in U$ there exists a local minorant $\varphi$
of $f$ of class $C^{2}$ that satisfies $\varphi(x_{0})=f(x_{0})$,
$D\varphi(x_{0})=Df(x_{0})$ and $D^{2}\varphi(x_{0})\in\mathcal{H}f(x_{0})$
(the generalized Hessian of $f$ at $x_{0}$). This question is motivated by the
second-order viscosity theory of the PDE, since for nonsmooth functions, an
analogous result between subgradients and first-order viscosity subjets is
known to hold in every separable Asplund space. In this work we show that the
aforementioned second--order result holds true whenever $\mathcal{H}f(x_{0})$
has a minimum with respect to the semidefinite cone (thus in particular, in
one dimension), but it fails in two dimensions even for piecewise polynomial functions.
We extend this result by introducing a new notion of directional minimum of $\mathcal{H}f(x_{0})$.