I'd like to ask questions about some steps in the following article, book.
- In the lemma 2.1 $ \int_{B_{\rho}} | w - w_{\rho}|^{2} \le \int_{B_{\rho}} | w - w(0)|^{2} $ and \begin{equation} \int_{B_\rho} | w - w(0)|^{2} \le \rho^{n+2} |Dw|_{L^{\infty}(B_{1/2}}^{2} \end{equation} As $w \in H^1(B_r), w$ could be redefined in $ 0$, could not?
Incidentally, is there a kind of inequality of the mean value for $w \in H^{1}(B_r)$?
2 In the theorem 2.4 we have \begin{equation} \lambda \int_{B_r(x_0)} |Dv|^{2} \le \int_{B_r(x_0)} (| a_{ij}(x_0) -a_{ij}(x))D_i u D_j v| + \int_{B_r(x_0)} |fv| \end{equation} Then, why \begin{equation} \int_{B_r(x_0)} |Dv|^{2} \le C\left \{ \tau^{2}(r) \int_{B_r(x_0)} |Du|^{2} + \Bigl( \int_{B_r(x_0)} |f|^{2n/(n+2)} \Bigr) \right \}? \end{equation} I know that \begin{eqnarray} \int_{B_r(x_0)} |Dv|^{2} \le \int_{B_r(x_0)} (| a_{ij}(x_0) -a_{ij}(x))D_i u D_j v| & \le & \int_{B_r(x_0)} \tau(r) |Du| |Dv| \end{eqnarray} Can I use Young's inequality whit $\varepsilon$ here?
And I know that \begin{eqnarray} \int_{B_r(x_0)} |fv| \le |v|_{2^{*}} |f|_{2^*/((2^* -1)} = |v|_{2^*} |f|_{2n/(n+2)} \end{eqnarray} But \begin{equation} |f|_{2n/(n+2)} = \Bigl( \int_{B_r(x_0} |f|^{2n/(n+2)}\Bigl)^{n+2/2n} \neq \Bigl( \int_{B_r(x_0} |f|^{2n/(n+2)}\Bigl)^{n+2/n} \end{equation} Finally. Why do we have to use Sobolev Inequality of the form \begin{equation} |v|_{2^*} \le C(n)|Dv|_{2}? \end{equation}