What is the importance of a Trace operator in PDE . Although I have read the Wiki page on this but I am not able to connect it to the aspect of solving PDE's.
Particularly why do we define Trace operator as $T\colon W^{1,p}(u) \to L^p(\partial U)$. Also why do we deal with only exponent $1$ . What is great about operator from $W^{1,p}$ to $L^p$?
Please help me to understand this concept of trace operator. Thank you very much .