• Value Functions for Certain Class of Hamilton Jacobi Equations

• # Fulltext

Permanent link:
https://www.ias.ac.in/article/fulltext/pmsc/121/03/0349-0367

• # Keywords

Hamilton Jacobi equation; viscosity solution; minimizing paths; value function.

• # Abstract

We consider a class of Hamilton Jacobi equations (in short, HJE) of type

$$u_t+\frac{1}{2}\left(|u_{x_n}|^2+\cdots+|u_{x_{n-1}}|^2\right)+\frac{e^u}{m}|u_{x_n}|^m=0,$$

in $\mathbb{R}^n\times\mathbb{R}_+$ and $m&gt;1$, with bounded, Lipschitz continuous initial data. We give a Hopf-Lax type representation for the value function and also characterize the set of minimizing paths. It is shown that the minimizing paths in the representation of value function need not be straight lines. Then we consider HJE with Hamiltonian decreasing in 𝑢 of type

$$u_t+H_1(u_{x_1},\ldots,u_{x_i})+e^{-u}H_2(u_{x_{i+1}},\ldots,u_{x_n})=0$$

where $H_1,H_2$ are convex, homogeneous of degree $n,m&gt;1$ respectively and the initial data is bounded, Lipschitz continuous. We prove that there exists a unique viscosity solution for this HJE in Lipschitz continuous class. We also give a representation formula for the value function.

• # Author Affiliations

1. TIFR Centre for Applicable Mathematics, Sharadanagar, Chikkabommasandra, Bangalore 560 065, India
2. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019

• # Proceedings – Mathematical Sciences | News

© 2017-2019 Indian Academy of Sciences, Bengaluru.