Let $\Omega$ be an open, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE?

$$|Du| = 1$$ $$\lim_{\delta \to 0_+} \text{sup}_{y, z \in B_d (x)} \,|Du(y) - Du(z)| = 0$$ $$u = 0 \text{ on } \partial \Omega.$$

Note: We say $u \in W^{1, \infty}$ is an almost everywhere solution if the first two equations above are satisfied for Lebesgue almost every $x \in \mathbb R^n$.

In other words, is there a Lipschitz solution to the Eikonal equation with almost everywhere continuous derivative?

Let $\Omega$ be an open, bounded, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE?

$$|Du| = 1$$ $$\lim_{\delta \to 0_+} \text{sup}_{y, z \in B_d (x)} \,|Du(y) - Du(z)| = 0$$ $$u = 0 \text{ on } \partial \Omega.$$

Note: We say $u \in W^{1, \infty}$ is an almost everywhere solution if the first two equations above are satisfied for Lebesgue almost every $x \in \mathbb R^n$.

In other words, is there a Lipschitz solution to the Eikonal equation with almost everywhere continuous derivative?

Let $\Omega$ be an open, connected subset of $\mathbb R^n$ with smooth boundary, and $f: \partial \Omega \to \mathbb R$ a Lipschitz continuous function. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following PDE?

$$\lim_{\delta \to 0_+} \text{sup}_{y, z \in B_d (x)} \,|Du(y) - Du(z)| = 0$$ $$u = f \text{ on } \partial \Omega.$$

Note: We say $u \in W^{1, \infty}$ is an almost everywhere solution if the first equation above is satisfied for Lebesgue almost every $x \in \mathbb R^n$.

In other words, is there a Lipschitz function with almost everywhere continuous derivative that agrees with $f$ on the boundary?

Let $\Omega$ be an open, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE?

$$|Du| = 1$$ $$\lim_{\delta \to 0_+} \text{sup}_{y, z \in B_d (x)} \,|Du(y) - Du(z)| = 0$$ $$u = 0 \text{ on } \partial \Omega.$$

Note: We say $u \in W^{1, \infty}$ is an almost everywhere solution if the first two equations above are satisfied for Lebesgue almost every $x \in \mathbb R^n$.

In other words, is there a Lipschitz solution to the Eikonal equation with almost everywhere continuous derivative?