Suppose is the open square in . is defined

Find the first weak derivative of .

Suppose is the weak derivative, then for any , we have

WLOG, assume . Let us denote the four domains in the definition of as respectively. Note that

While

obtained from the integral by parts and the boundary behavior of . Similarly for domain

As for the domain and , we have

Substituting to

.

So from , we know

As for , one can get it in a similar way.

Evans, partial differential equation. 5.3

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