I need to prepare a small topic from Evan's Partial Differential Equations. There are a few things I wasn't able to understand so far. I hope you can help me with these things: Evans defines which requirements a weak solution of the Hamilton-Jacobi Equation has to fulfill and then goes on to proof it's uniqueness:
In his proof he claims inequality (43) to be true, "Verification is left as an exercise":
But I'm not able to understand why inequality (c) should imply (43). I guess you have to use that the derivative of a convolution equals the convolution of function $u$ and the derivative of $\eta_\varepsilon$.
Furthermore I cannot see how inequality (46) is implied by (41) and (43):
Isn't $b_\varepsilon$ a scalar field?
I would be very happy if anyone could me, I'm really confused right now.