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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 .

1 Answers 1

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a) If $u$ is defined on $B_R(x)$, consider the affine map $Ty=Ry+x$. The composition $u\circ T$ is defined on $B_1(0)$. Apply the known inequality to it. Use the Chain Rule to calculate $\nabla (u\circ T)$. Use the change-of-variables formula to move the integrals back to $B_R(x)$.

b) Expand the integral of $|u-s|^2$ as a quadratic polynomial of $s$, and look for its minimum.

c) will be easy once you understand a) and b).

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