The following argument is made in Evans's Partial Differential Equations, Chapter 5.3.3:
Here are my questions:
- As I can read from the argument, $\lambda\varepsilon$ would be eventually small enough if $\varepsilon>0$ is small enough. Why on earth would one need $\lambda$ to be sufficiently large? What if we just define $$ x^\varepsilon=x+\varepsilon e_n? $$
- What does "there is room to mollify within $U$" mean? How does one know $$ v^\varepsilon\in C(\overline{V})? $$