Herglotz’ Generalized Variational Principle and Contact Type Hamilton-Jacobi Equations

Abstract

We develop an approach for the analysis of fundamental solutions to Hamilton-Jacobi equations of contact type based on a generalized variational principle proposed by Gustav Herglotz. We also give a quantitative Lipschitz estimate on the associated minimizers.

Appendix

Let $\varOmega \subset \mathbb {R}^{n+1}$ be an open set. A function $f:\varOmega \subset \mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ is said to satisfy Carathéodory condition if

• for any $x\in \mathbb {R}^n$, $f(\cdot ,x)$ is measurable;

• for any $t\in \mathbb {R}$, $f(t,\cdot )$ is continuous;

• for each compact set U of $\varOmega$, there is an integrable function $m_U(t)$ such that

  $$ |f(t,x)|\leqslant m_U(t),\quad (t,x)\in U. $$

A classical problem is to find an absolutely continuous function x defined on a real interval I such that $(t,x(t))\in \varOmega$ for $t\in I$ and satisfies the following Carathéodory equation

$$\begin{aligned} \dot{x}(t)=f(t,x(t)),\quad a.e., t\in I. \end{aligned}$$

(41)

Proposition 8

(Carathéodory) If $\varOmega$ is an open set in $\mathbb {R}^{n+1}$ and f satisfies the Carathéodory conditions on $\varOmega$, then, for any $(t_0, x_0)$ in $\varOmega$, there is a solution of (41) through $(t_0, x_0)$. Moreover, if the function f(t, x) is also locally Lipschitzian in x with a measurable Lipschitz function, then the uniqueness property of the solution remains valid.

For the proof of Proposition 8 and more results related to Carathéodory equation (41), the readers can refer to [17, 24].