**Problem: **Derive the product formula for weak derivative

holds for all such that and .

*Proof:* Step 1: prove the case ,

Step 2: prove the case by step 1. Readers can also see the proof on page269 of Functional Analysis, Sobolev Spaces and Partial Differential Equations (Haim Brezis)

Step 3: Define

then is piecewise smooth in and . So by lemma 7.8 in Gilbarg and Trudinger’s bk and

Denote and . By step 2 we have and

Note that by assumption

By dominating convergence theorem, letting

Step 4:Consider the problem with additional assumption and .

Firstly we assume . Define . Since , then , satisfies , moreover

Suppose defined on , where . Then is a piecewise smooth function in and . By lemma 7.8 on Gilbarg and trudinger’s book, . Note that by the conclusion of step 3,

that is

By dominating convergence theorem, as

And

Letting , implies

Since for any

by dominating convergence theorem, letting

If we only know , consider and , repeat the above proof

note that , this is equivalent to

Step 5: Consider the most general case with no extra assumption. Since

step 4 will imply

So

and

**Remark:** Who can simplify this proof? It is ugly.

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