Barrier functions for Pucci-Heisenberg operators and applications

The aim of this article is the explicit construction of some barrier functions ("fundamental solutions") for the Pucci-Heisenberg operators. Using these functions we obtain the continuity property, up to the boundary, for the viscosity solution of fully non-linear Dirichlet problems on the Heisenberg group, if the boundary of the domain satisfies some regularity geometrical assumptions (e.g. an exterior Heisenberg-ball condition at the characteristic points). We point out that the knowledge of the fundamental solutions allows also to obtain qualitative properties of Hadamard, Liouville and Harnack type

The ergodic problem for some subelliptic operators with unbounded coefficients

We study existence and uniqueness of the invariant measure for a stochastic process with degenerate diffusion, whose infinitesimal generator is a linear subelliptic operator in the whole space R N with coefficients that may be unbounded. Such a measure together with a Liouville-type theorem will play a crucial role in two applications: the ergodic problem studied through stationary problems with vanishing discount and the long time behavior of the solution to a parabolic Cauchy problem. In both cases, the constants will be characterized in terms of the invariant measure

Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface $\Gamma_\epsilon$. The oscillations of the interface have small period and amplitude, both of the order of $\epsilon$, and the interfaces $\Gamma_\epsilon$ tend to a straight line $\Gamma$. We study the asymptotic behavior as $\epsilon\to 0$. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $\Gamma$, with an effective transmission condition on $\Gamma$ keeping track of the oscillations of $\Gamma_\epsilon$

Trace Results on Domains with Self-Similar Fractal Boundaries

International audienceThis work deals with trace theorems for a family of ramified domains $\Omega$ with a self-similar fractal boundary $\Gamma^\infty$. The fractal boundary $\Gamma^\infty$ is supplied with a probability measure $\mu$ called the self-similar measure. Emphasis is put on the case when the domain is not a $\epsilon-\delta$ domain and the fractal is not post-critically finite, for which classical results cannot be used. It is proven that the trace of a square integrable function belongs to $L^p_\mu$ for all real numbers $p\ge 1$. A counterexample shows that the trace of a function in $H^1(\Omega)$ may not belong to $BMO(\mu)$ (and therefore may not belong to $L^\infty_\mu$). Finally, it is proven that the traces of the functions in $H^1(\Omega)$ belong to $H^s(\Gamma^\infty)$ for all real numbers $s$ such that $0\le s d_H/4$ are supplied. \\ There is an important contrast with the case when $\Gamma^\infty$ is post-critically finite, for which the square integrable functions have their traces in $H^s(\Gamma^\infty)$ for all $s$ such that $0\le

Neumann conditions on fractal boundaries

We consider some elliptic boundary value problems in a self-similar ramified domain of ℝ2 with a fractal boundary with Laplace's equation and nonhomogeneous Neumann boundary conditions. The Hausdorff dimension of the fractal boundary is greater than one. The goal is twofold: first rigorously define the boundary value problems, second approximate the restriction of the solutions to subdomains obtained by stopping the geometric construction after a finite number of steps. For the first task, a key step is the definition of a trace operator. For the second task, a multiscale strategy based on transparent boundary conditions and on a wavelet expansion of the Neumann datum is proposed, following an idea contained in a previous work by the same authors. Error estimates are given and numerical results are presented

Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction

We consider a family of open star-shaped domains made of a finite number of non intersecting semi-infinite strips of small thickness and of a central region whose diameter is of the same order of thickness, that may be called the junction. When the thickness tends to 0, the domains tend to a union of half-lines sharing an endpoint. This set is termed "network". We study infinite horizon optimal control problems in which the state is constrained to remain in the star-shaped domains. In the above mentioned strips the running cost may have a fast variation w.r.t. the transverse coordinate. When the thickness tends to 0 we prove that the value function tends to the solution of a Hamilton-Jacobi equation on the network, which may also be related to an optimal control problem. One difficulty is to find the transmission condition at the junction node in the limit problem. For passing to the limit, we use the method of the perturbed test-functions of Evans, which requires constructing suitable correctors. This is another difficulty since the domain is unbounded

Hamilton–Jacobi equations for optimal control on junctions and networks

erratum de l'article dans ESAIM COCV, 2016, vol. 22 n° 2, pp. 539-542 ;doi : 10.1051/cocv/2016005We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. A notion of viscosity solution of Hamilton-Jacobi equations on the network has been proposed in earlier articles. Here, we propose a simple proof of a comparison principle based on arguments from the theory of optimal control. We also discuss stability of viscosity solutions. Résumé. On consid ere desprobì emes de contrôle optimal pour lesquels l'´ etat est contraint a rester sur un réseau. Une notion de solution de viscosité des equations de Hamilton-Jacobi associées a ´ eté proposée dans des travaux antérieurs. Ici, on propose une preuve simple d'un principe de comparaison. Cette preuve est basée sur des arguments de contrôle optimal. La stabilité des solutions de viscosité est aussí etudiée

JLip versus Sobolev Spaces on a Class of Self-Similar Fractal Foliages

International audienceFor a class of self-similar sets $\Gamma^\infty$ in $\R^2$, supplied with a probability measure $\mu$ called the self-similar measure, we investigate if the $B_s^{q,q}(\Gamma^\infty)$ regularity of a function can be characterized using the coefficients of its expansion in the Haar wavelet basis. Using the the Lipschitz spaces with jumps recently introduced by Jonsson, the question can be rephrased: when does $B_s^{q,q}(\Gamma^\infty)$ coincide with $JLip(s,q,q;0;\Gamma^\infty)$? When $\Gamma^\infty$ is totally disconnected, this question has been positively answered by Jonsson for all $s,q$, $00$, $1\le p,q<\infty$, using possibly higher degree Haar wavelets coefficients). Here, we fully answer the question in the case when $

Comparison of Different Definitions of Traces for a Class of Ramified Domains with Self-Similar Fractal Boundaries

International audienceWe consider a class of ramified bidimensional domains with a self-similar boundary, which is supplied with the self-similar probability measure. Emphasis is put on the case when the domain is not an epsilon-delta domain as defined by Jones and the fractal is not totally disconnected.We compare two notions of trace on the fractal boundary for functions in some Sobolev space, the classical one ( the strict definition ) and another one proposed in 2007 and heavily relying on self-similarity. We prove that the two traces coincide almost everywhere with respect to the self similar probability measure