Uniqueness of weak solutions of HAMILTON-JACOBI EQUATIONS (Evans PDE, P131)

Question

In the proof of THEOREM 7 (Uniqueness of weak solution) in section 3.3.3 of the book "Partial differential equations" written by Lawrence C. Evans, there's a claim How you get equations (41), (42) and (43) is beyond me. I checked $\S$C.4, but only found the following listed

Consider a family of smooth, bonded regions $U(\tau)\subset\mathbb{R}^n$ that depend smoothly upon the parameter $\tau\in\mathbb{R}$. Write ${\bf v}$ for the velocity of the moving boundary $\partial U(\tau)$ and $\nu$ for the outward pointing unit normal.

${\bf THEOREM 6}$ (Differentiation formula for moving regions). If $f=f(x,\tau)$ is a smooth function, then $$\frac{d}{d\tau}\int_{U(\tau)}fdx=\int_{\partial U(\tau)}f\ {\bf v} \cdot\nu dS+\int_{U(\tau)}f_{\tau}dx.$$

How do we derive from this?

Answer 1

This is an error as noted in the errata for the second edition, maintained on Evans' homepage:

Chapter 3

[...] page 132, line 5: Change “C.4” to “C.5”

C.5 is about mollification; in particular you will want Theorem 7 of C.5.