[Picture] [Picture] 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! [Picture] [Picture] [Picture] [Picture] 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. [Picture] Next we summarise the three approximation theorems and prove also the second one. [Picture] [Picture] 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: [Picture] [Picture] A function in W^1,p defines an L^p function on the boundary if the domain is bounded and the boundary is smooth: [Picture] So the L^p norm of the restriction of u to the boundary is controled by W^1,p norm of u. [Picture] 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.