Consider $r>0$ , $1<p< \infty $. Consider $K $ a compact set in $R^n$ with $K \subset B(x_0 , r)$.
Define
$$ B = \{ u \in C^{\infty}_{0}(B(x_0 , r)) ; u=1 \ on \ K , 0 \leq u \leq 1\}$$
$$ A= \{ u \in C^{\infty}_{0}(B(x_0 , r)) ; u=1 \ \textit{in a open neighborhood of K } \}$$
A natural question is :
given $u \in B$ , exists a sequence $v_n \subset A$ with $v_n \rightarrow u$ in $H^{1,p}( B(x_0 , r))$ ?
I tried to prove this fact , but nothing... I draw a picture and I belive the result is true. Someone can give me a help ?If i prove the affirmation , then I understand a theorem in the book that i am studying.
thanks in advance =)