a question about chapter9 in EVANS pde

Question

I'm reading chapter 8 of the pde book written by EVANS. It's about variation method of pde. In page 433. what's the mean of $C_c^\infty(U)$? why the test function $v$ is in this space?

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thank you!

Answer 1

$C_{c}^{\infty}(U)$ denote the space of infinitely differentiable functions with compact support in $U$. By definition, the function belonging to this space called a test function. See page 242 of Evans's book for more details.