proof of the Meyers-Serrin Theorem in Evans's PDE book

Question

The following is the Meyers-Serrin theorem and its proof in Evans's Partial Differential Equations:

Could anyone explain where (for which $x\in U$) is the convolution in step 2 defined and how to get (3) from Theorem 1?

Answer 1

The following theorem is from Adam's Sobolev Spaces

If $u\in L^1_{\hbox{loc}}(\Omega)$ and $\text{supp }(u)\subset\subset\Omega$, then $\eta_\varepsilon*u\in C_c^\infty(\Omega)$ provided $$ \varepsilon<\text{dist }(\text{supp }(u),\partial U). $$

Also, we have the mollification in $W^{k,p}(\Omega)$ which is related to Theorem 1 in OP:

Let $1\leq p<\infty$ and $u\in W^{k,p}(\Omega)$. If $\Omega'$ is a subdomain with compact closure in $\Omega$, then $\eta_\varepsilon*u\to u$ in $W^{k,p}(\Omega')$.

Because we have $\zeta_iu\in W^{k,p}(U)$ (thus in $L^1_{\hbox{loc}}(U)$), and $\text{spt }(\zeta_iu)\subset V_i\subset\subset U$, the convolution $u^i=\eta_{\varepsilon_i}*(\zeta_iu)$ is defined (for small enough $\varepsilon_i>0$) everywhere in $U$, comparing to the $f^\varepsilon$ in Theorem 1 that it is only well-defined in $U_\varepsilon$.

One can (for example done in Adam's book, Thm 3.17, which is the same as Theorem 2 in OP) explicitly calculate the $\varepsilon_i>0$ such that $$ \text{spt }u^i\subset W_i $$ and $$ \|u^i-(\zeta_iu)\|_{W^{k,p}({W_i})}\leq\delta'\qquad\text{by the second theorem quoted from Adam.} $$

Since both $u^i$ and $\zeta_iu$ are in $W^{k,p}(U)$ (noting that $u^i\in C^\infty_c(U)$), and both have support in $W_i$, one has $$ \|u^i-(\zeta_iu)\|_{W^{k,p}(U)}=\|u^i-(\zeta_iu)\|_{W^{k,p}({W_i})}. $$ (3) is thus established.

Answer 2

In Evans' book, $\eta_\varepsilon$ is the standard mollifiers, and in my edition is defined/discussed in appendix C.4.

Here was my thought process in thinking about your question:

Looking at the definition in the appendix, $$f^\varepsilon(x)=(f*\eta_\varepsilon)(x)=\int_{B(0,\varepsilon)}\eta_\varepsilon(y)f(x-y)dy=\int_{B(x,\varepsilon)}\eta_\varepsilon(x-y)f(y)dy.$$ We are essentially taking the mollifier $\eta_\varepsilon$, whose support sits inside the ball $B(0,\varepsilon)$, and moving it around $U$ so that we can weight $f$ by it at each point $x$ with the goal of smoothing out $f$. Thus in order for the integrad $\eta_\varepsilon(y)f(x-y)$ to even make sense, the ball sitting at $x$ has to be completely inside $U$. This is motivation for the notation/definition $U_\varepsilon=\{x\in U \ | \ \mathrm{dist}(x,\partial U)<\varepsilon\}$, and why the (basically) direct proof of Theorem 1 only gives local approximation, and why "dealing with the boundary" is the main goal of this result.

Thus, the convolutions $u^i$ are defined for any $x\in U_{\varepsilon_i}$. Theorem 1 then implies that, since $\zeta_iu\in W^{k,p}(U)$, we have a local approximation of smooth functions, but since $\mathrm{spt}(\zeta_iu)\subset V_i\subset\subset U$, we may presumably take the $\varepsilon_i$'s small enough that the convolution is defined on all of $V_i$ so we may drop the "$\mathrm{loc}$" from the approximation. This gives us the sequence $u^i$ satisfying $(3)$ without loss of generality.

Hope this helped clarify the proof!