Semicontinuous viscosity solutions to mixed boundary value problems with degenerate convex Hamiltonians
This paper is devoted to the study of a boundary value problem of the form \[ \begin{cases} H(x,Du,u_k)+ \lambda u=0\quad &\text{in }\Omega\times (0, \infty),\\ H(x,Du,u_k)+ \lambda u=0\quad &\text{or }u=g\text{ on }\partial\Omega \times (0,\infty),\\ H(x,Du,u_k) +\lambda u\geq 0\quad &\text{on }\Omega\times \{0\},\end{cases} \tag{1} \] where \(\Omega\) is an open subset of \(\mathbb{R}^n\), and \(g\) is a bounded lower semicontinuous function. Here the authors prove the uniqueness of the solution to (1) in the viscosity sense, and show its existence by giving a representation formula in terms of the value function of an exit time control problem.
scientific article; zbMATH DE number 1751605
scientific article