Regularity of the Viscosity Solution of a Kind of Hamilton-Jacobi Equations

Fátima Pereira^* and Vladimir Goncharov

We consider a simple (stationary) \emph{Hamilton-Jacobi equation} \begin{equation} \rho _{F}\left( \nabla u\left( x\right) \right) =1\text{,} \tag{$\star$} \end{equation} where $F$ is a closed convex bounded subset of a Hilbert space $H$ such that $0\in \mathrm{int}\,F$, and $\rho _{F}\left( \cdot \right) $ stands for the \emph{Minkowski functional} associated to $F$. It is well-known that the unique \emph{viscosity solution} of $(\star)$ on an open set $\Omega \subset H$ with the boundary datum $u\left\vert _{\partial \Omega }\right. =\theta $ is given by the formula: \[ \hat{u}\left( x\right) =\underset{y\in C}{\inf }\left\{ \theta \left( y\right) +\rho _{F^{0}}\left( x-y\right) \right\} \text{,} \] where $C:=H\setminus \Omega $ and $F^{0}$ is the polar set to $F$, as soon as the function $\theta $ satisfies a natural \emph{slope condition}. In the talk we propose some geometric hypotheses guaranteeing the regularity of $% \hat{u}\left( \cdot \right) $ (the \emph{Fréchet differentiability} and the (\emph{Hölder}) \emph{continuity} of its gradient) near the boundary $\partial \Omega $. The obtained results are illustrated by some simple examples in $\mathbb{R}^{2}$.

Mathematics Subject Classification: 49L25 49J52 49N15

Keywords: time-minimum problem; viscosity solution; duality mapping; proximal normals; proximal regularity; Hölder continuity.

Minisymposion: Dynamic Programming Approach to Optimal Control Methods and Applications