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\title{Hamitlon-Jacobi equations on networks}
\author{Fabio Camilli}
\institute{``Sapienza'' Universit{\`a}  di Roma}
\email{camilli@dmmm.uniroma1.it}
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\section{Abstract}
I will provide a survey of recent results on the study of Hamilton-Jacobi equations on networks.
Note that the problem  is not a straightforward generalization
of the Euclidean setting  (the one thoroughly studied, e.g., in the
Bardi-Capuzzo Dolcetta book \cite{bcd})  because of the presence
of the vertices. \par
With the aim  of extending    the notion   of viscosity solution to   networks,  three different approaches      have been recently proposed (\cite{acct}, \cite{imz}, \cite{sc}).
All the three papers  lead to a mathematically
convenient theory (i.e. a  theory  for which  existence, uniqueness  and stability hold).  But since they  are motivated by different applications (respectively, a control problem constrained to a network in \cite{acct}, the study of traffic flow at a junction in \cite{imz} and Eikonal equations and  distance functions on  networks in \cite{sc}), they differ  for the assumptions made on the Hamiltonian and especially for the definitions of viscosity solution at the vertices (while inside the edges all the definitions coincide with the classical one). \par
In this talk I will mainly concentrate on  the results in \cite{sc}, but I will also try to point out    similarities and differences with the  other two approaches.\par
 A preliminary  plan of the talk  is the following:\\
In the $1^{st}$ lecture, I will briefly  review the linear  theory of differential equations on networks. In the  $2^{nd}$  lecture I will describe the notion of viscosity solution for Hamilton-Jacobi equation on networks in \cite{sc}. The  $3^{rd}$ lecture is devoted to the comparison among the various notions of viscosity solution.  In the last 45 minutes-lecture I will switch to  a different, even if connected argument, namely eikonal equation on  Sierpinski gasket and pcf fractal sets.


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\bibitem{acct} \sc{Y. Achdou, F. Camilli, A. Cutr\`i and  N. Tchou}, \emph{Hamilton-Jacobi equations constrained on networks},  NoDEA Nonlinear Differential Equations Appl. 20  (2013), 413--445.

\bibitem{bcd}
\sc{M.~Bardi and I.~Capuzzo~Dolcetta},
  \emph { Optimal control and viscosity solutions of {H}amilton-{J}acobi-{B}ellman equations}.
\ Birkh\"auser Boston Inc., Boston, MA, 1997.

\bibitem{imz}
 {\sc C. Imbert, R. Monneau and H. Zidani}, \emph{A Hamilton-Jacobi approach to junction problems and application to traffic flows},  ESAIM Control Optim. Calc. Var. 19 (2013), 129-166.

\bibitem{sc}
{\sc D. Schieborn and F. Camilli},\emph{ Viscosity solutions of Eikonal equations on
topological network},   Calc. Var. Partial Differential Equations. 46 (2013), 671--686.

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