For $\phi\in C^1(\mathbb{R}^N)$ with $$\omega_{\phi}=\{x\in\mathbb{R}^N\ |\ \phi(x)>0\}$$ being a bounded set with $\nabla\phi (x)\neq 0,\ \forall\ x\in\phi^{-1}(0)=\partial\omega_{\phi}\neq \emptyset$ consider the **signed distance function**:

$$ d_{\phi}:\mathbb{R}^N\to\mathbb{R},\ d_{\phi}(x)=\begin{cases} -\text{dist}(x,\partial\omega_{\phi}),\ x\in\omega_{\phi} \\ 0,\ x\in\partial\omega_{\phi} \\ \text{dist}(x,\partial\omega_{\phi}),\ x\in\mathbb{R}^N\setminus\overline{\omega}_{\phi}\end{cases}$$

**I have some technical questions that seem to be true about $d_{\phi}$**.

Let's take a point $x_0\in\mathbb{R}^N$ with a unique $y(0)\in\partial\omega_{\phi}$ such that: $|x_0-y|=\text{dist}(x_0,\partial\omega_{\phi})$ and any $\psi\in C^1(\mathbb{R}^N)$ with $|\psi(x)|<M,\forall\ x\in\mathbb{R}^N$.

For $|\theta|$ sufficiently small we associate to $x_0$ a set denoted $Y(\theta)$ such that $\forall y\in Y(\theta)\subseteq\mathbb{R}^N$ we have $\phi(y)+\theta\psi(y)=0$ and:

$$d_{\phi+\theta\psi}(x_0)=|x_0-y|.$$

How can we prove that:

**(1)** There is a selection for the multifunction $Y:[0;\theta_0)\rightrightarrows \phi^{-1}(0)$ denoted $y=y(\theta)$ which is continuous and differentiable for $\theta\in [0,\theta_0)$.

Note that we cannot apply here Michael's Selection Theorem because in general $Y(\theta)$ is not a convex set. (https://en.wikipedia.org/wiki/Michael_selection_theorem)

**(2)** $\lim\limits_{\theta\to 0} d_{\phi+\theta\psi}(x_0)=d_{\phi}(x_0)$

**(3)** Exists $\lim\limits_{\theta\to 0}\dfrac{d_{\phi+\theta\psi}(x_0)-d_{\phi}(x_0)}{\theta}$

I tried a lot to prove the continuity but I did not succeed.

It is well known (from KKT conditions) that $y(0)$ and $y(\theta)$ satisfy:

$$\begin{cases} \phi(y(0))=0,\ \phi(y(\theta))+\theta\psi(y(\theta))=0 \\ \lambda\nabla \phi (y(0))=y(0)-x_0,\ \lambda_{\theta}\Big (\nabla\phi(y(\theta))+\theta\psi(y(\theta))\Big )=y(\theta)-x_0\end{cases} $$

I proved easily that for $\theta$ sufficiently small $\nabla\phi+\theta\nabla\psi\neq 0$ on $(\phi+\theta\psi)^{-1}(0)\neq \emptyset$. Also note that if we prove $(1)$ then the answer to $(2)$ follows immediately and $(3)$ can be computed from KKT conditions given the result: $-\dfrac{\psi(y(0))}{|\nabla\phi(y(0))|}$.

Maybe the following sources will help:

Set of points with a unique closest point in a compact set

Concavity near the boundary of the distance function

Finally, are there some references that treats the signed distance function with the level set method (not with a shape derivative approach, but a functional approach)?