Well-posedness for constrained Hamilton-Jacobi equations
2018
The goal of this paper is to study a Hamilton-Jacobi equation \begin{equation*}
\begin{cases}
u_t=H(Du)+R(x,I(t)) &\text{in }\mathbb{R}^n \times (0,\infty),
\sup_{\mathbb{R}^n} u(\cdot,t)=0 &\text{on }[0,\infty),
\end{cases} \end{equation*} with initial conditions $I(0)=0$, $u_0(x,0)=u_0(x)$ on $\mathbb{R}^n$. Here $(u,I)$ is a pair of unknowns and the Hamiltonian $H$ and the reaction $R$ are given. And $I(t)$ is an unknown constraint (Lagrange multiplier) that forces supremum of $u$ to be always zero. We construct a solution in the viscosity setting using a fixed point argument when the reaction term $R(x,I)$ is strictly decreasing in $I$. We also discuss both uniqueness and nonuniqueness. For uniqueness, a certain structural assumption on $R(x,I)$ is needed. We also provide an example with infinitely many solutions when the reaction term is not strictly decreasing in $I$.
Keywords:
- Correction
- Source
- Cite
- Save
- Machine Reading By IdeaReader
11
References
0
Citations
NaN
KQI