So u_eps is smooth and the mollified weak derivative
is the derivative of the mollified function. Interior approximation
in W^1,p then only relies on Theorem 6 (iv) in App C.4,
which we discuss below. N.B. p is finite!
So an L^1_loc function is zero a.e. if it kills all test functions.
Below we recall what this implies: we can integrate the weak
derivative, as is done in the basic GNS and Morrey estimates.
Next we summarise the three approximation theorems
and prove also the second one.
A function in W^1,p may be extended to a W^1,p function supported
in a slightly larger domain if the domain is bounded and the boundary
is smooth:
A function in W^1,p defines an L^p function on the boundary
if the domain is bounded and the boundary
is smooth:
So the L^p norm of the restriction of u to the boundary is controled
by W^1,p norm of u.
Idea of proof of Rellich-Kondrachov compactness theorem:
Only need to do q=1, in view of App B.2.h.
Extend all u_m to a slightly larger domain. The mollified
u_m^eps are close to u_m uniformly in m if eps is small.
For fixed eps they are smooth, with unform bounds on u_m^eps
and its gradient. Therefore there is for each eps a
uniformly convergent subsequence, which is certainly convergent
in L^1. Take a sequence of epsilons converging to zero
and apply a diagonal argument to the resulting diagramm
of subsequences of subsequences to conclude.