I have the following question:

Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that

1. $F(x,\cdot)$ is convex with respect to the second variable.

2. $F(\cdot,v)$ is upper (or lower) semicontinuous with respect to the first variable.

3. $F(x,tv)=|t|F(x,v)$ and $F(x,v)>0$ unless $v=0$.

4. $F(x,\cdot)$ is elliptic in the sense that there exists continuous function $g:\Omega\to[1,\infty)$ such that $$g(x)^{－1}|v|\leq F(x,v)\leq g(x)|v|$$ for all $x\in \Omega$ and $v\in \mathbb{R}^n$.

Consider the following equation, $u\in C^\infty(\Omega)$

\begin{equation} (\frac{d}{dt}u(\gamma(t)):=)<\nabla u(\gamma(t)),\gamma'(t)>=F(\gamma(t),\gamma'(t))\\ \gamma(0)=x_0. \end{equation}

Does the above Halmiton-Jacobian equation admits a Lipschitz solution for short time with any given initial data $x_0\in \Omega$, namely the above equation admits a Lipschitz curve $\gamma$?

Any suggestion, comments are welcome.

Note: since the condition on $F$ is so weak, it might be the case that there is no solution to this equation at all!

I have the following question:

Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that

1. $F(x,\cdot)$ is convex with respect to the second variable.

2. $F(\cdot,v)$ is upper (or lower) semicontinuous with respect to the first variable.

3. $F(x,tv)=|t|F(x,v)$ and $F(x,v)>0$ unless $v=0$.

4. $F(x,\cdot)$ is elliptic in the sense that there exists continuous function $g:\Omega\to[1,\infty)$ such that $$g(x)^{－1}|v|\leq F(x,v)\leq g(x)|v|$$ for all $x\in \Omega$ and $v\in \mathbb{R}^n$.

Consider the following equation, $u\in C^\infty(\Omega)$

\begin{equation} (\frac{d}{dt}u(\gamma(t)):=)\langle\nabla u(\gamma(t)),\gamma'(t)\rangle=F(\gamma(t),\gamma'(t))\\ \gamma(0)=x_0. \end{equation}

Does the above Halmilton-Jacobian equation admit a Lipschitz solution for short time with any given initial data $x_0\in \Omega$? That is, can the above equation admit a Lipschitz curve $\gamma$ as a solution?

Any suggestion or comments are welcome.

Note: since the condition on $F$ is so weak, it might be the case that there is no solution to this equation at all!