Derivate of distance function and inner normal of boundary of a domain

\mathbf{Problem:} Let  \Omega\subset \mathbb{R}^n has non-empty boundary \partial \Omega\in C^2. Let \nu(y)denote the unit inner normal to \partial \Omega at y. WLOG assume \partial \Omega is given by x_n=\phi(x') where x'=(x_1,x_2,\cdots,x_{n-1}) and \psi\in C^2(\mathcal{N}\cap{x_n=0}) and D\phi(y'_0)=0, where \mathcal{N} is a neighborhood of y_0.  Also assume \nu(y_0) is the x_n coordinate axis.

The unit inner normal vector \nu(y) at a point y=(y',\psi(y'))\in \mathcal{N}\cap \partial \Omega is given by

\displaystyle \nu_i(y)=\frac{-D_i\psi(y')}{\sqrt{1+|D\psi(y')|^2}} i=1,2,\cdots,n-1. \quad \nu_n(y)=\displaystyle \frac{1}{\sqrt{1+|D\psi(y')|^2}}\quad (1)

Since \partial \Omega\in C^2, there exists a neighborhood of \partial \Omega, say \chi_\epsilon=\{x\in \overline{\Omega}|d(x)<\epsilon\} such that for \forall\, x\in\chi_\epsilon, there will exist a unique point y=y(x)\in\partial \Omega such that |x-y|=d(x). Then x,y satisfy

x=y+\mathbf{\nu}(y)d\quad (2)

Prove that for each point x\in \chi_\epsilon, we have Dd(x)=\nu(y(x)).

\mathbf{Proof:} Since |\nu(y)|=1, (2) means d=(x-y)\cdot\nu(y). Then y=y(x) is C^1(see GT’s book p355).

\displaystyle d=\sum\limits_{j=1}^n(x_j-y_j)\nu_j(y), y_n=\phi(y') and \nu(y)=\nu(y',y_n) is given by (1)

\displaystyle D_kd(x)=\nu_k(y)+\sum\limits_{j,\,l=1}^nx_jD_l\nu_j D_k y_l-\sum\limits_{j=1}^nD_ky_j\nu_j(y)-\sum\limits_{j,\,l=1}^ny_jD_l\nu_j(y)D_ky_l

\displaystyle =\nu_k(y)+\sum\limits_{j,\,l=1}^n(x_j-y_j)D_l\nu_j D_k y_l-\sum\limits_{j=1}^nD_ky_j\nu_j(y)

\displaystyle =\nu_k(y)+d\sum\limits_{j,\,l=1}^n\nu_jD_l\nu_j D_k y_l-\sum\limits_{j=1}^nD_ky_j\cdot\nu_j(y)

Actually the last two terms are equal to 0. In fact.

\displaystyle \sum\limits_{j,\,l=1}^n\nu_jD_l\nu_j D_k y_l=D_k|\nu(y)|^2=0 because |v|=1

Using (1)

\displaystyle \sum\limits_{j=1}^nD_ky_j\cdot\nu_j(y)=\sum\limits_{i=1}^{n-1}\frac{-D_i\psi(y')D_ky_i}{\sqrt{1+|D\psi(y')|^2}}+\frac{\sum_j D_j\psi(y')D_ky_j}{\sqrt{1+|D\psi(y')|^2}}=0

\text{Q.E.D}\hfill \square

\mathbf{Remark:} GT’s book p355. Chapter 14. Boundary curvatures and Distance Function.

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