Given $\int_{B_1 (0)}|u|^2dx \le C \int_{B_1 (0)} |\triangledown u|^2 dx$, where $C \in (0, \infty ) $, $B_1(0) \subset R^d $ and $u \in H_0^1(B_1(0))$. I have few questions ,
a) how can i via scaling , shifting , find a general inequality for balls $B_R(x)$
b)Let $u \in H^1(B_R(x_0))$, how can i find $s_\star \in R$ such that for every $s \in R$ we have $\int_{B_R (x_0)}|u(x)-s_\star|^2dx \le \int_{B_R (x_0)} |\ u(x)-s|^2 dx$.
c) How can i show that there exists $C=C(n)$ such that for every $u \in H^1(B_R(x_0))$ we have $\int_{B_R (x_0)}|u(x)-[u]_{B_R(x_0)}|^2dx \le CR^2\int_{B_R (x_0)} |\triangledown u|^2 dx$, where $[u]_A=\frac1{|A|}\int_A (u(x)) dx$ .
I don't know how to approach to this problem . Can anybody help me out . If possible please do tell me what exactly i should look for to solve this problem. Thank you .