Variational Strong Maximum Principle and Convolutions

Let us given a closed convex bounded set $F\subset \mathbb{R}^{n}$, containing the origin in its interior, a closed set $C\subset \mathbb{R}^{n}$ and a function $\theta \left( \cdot \right)$ satisfying the \emph{slope condition} \begin{equation*} \theta \left( x\right) -\theta \left( y\right) \leq \rho _{F^{0}}\left( x-y\right) \quad \forall x,y\in C\text{,} \end{equation*} where $F^{0}$ is the \emph{polar set} to $F$ and $\rho _{F}\left( \cdot \right)$ is the \emph{Minkowski functional} associated to $F^{0}$. In the talk we present a property of the \emph{convolution} $$\hat{u}\left( x\right) =\underset{y\in C}{\inf }\left\{ \theta \left( y\right) +\rho _{F^{0}}\left( x-y\right) \right\} \tag{\star}$$ regarding to the \emph{Strong Maximum Principle} (in the variational setting) for the integral functional \begin{equation*} \underset{\Omega }{\int }f\left( \rho _{F}\left( \nabla u\left( x\right) \right) \right) \,dx\text{,} \end{equation*} where $\Omega \subset \mathbb{R}^{n}$ is an open bounded connected domain such that $C\subset \Omega$, and $f:\mathbb{R}^{+}\rightarrow \mathbb{R} ^{+}\cup \left\{ +\infty \right\}$ is a convex lower semicontinuous function with $\ f\left( 0\right) =0$. Observe that, on the other hand, the function $(\star)$ can be interpreted as the (unique) \emph{viscosity solution} of the \emph{Hamilton-Jacobi equation} \begin{equation*} \rho _{F}\left( \nabla u\left( x\right) \right) =1\text{,}\quad u\left\vert _{\partial \Omega }\right. =\theta \text{,} \end{equation*} on the (open) set $\Omega \setminus C$ with the (appropriately extended) boundary function $\theta \left( \cdot \right)$.