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I am reading the book Elliptic and Parabolic Equations and the proof is excerpted from page 133-136.

  1. In Theorem 5.1.3: enter image description here

    it claims that $$C((1-\theta)R)^{-n/2}\left(\sup_{B_R}u\right)^{1-p/2}\left(\int_{B_R}u^p\,dx\right)^{1/2}\le\frac{1}{2}\sup_{B_R}u+C((1-\theta)R)^{-n/p}\|u\|_{L^p(B_R)}$$ , according to Young's inequality with ε. But I cannot figure out what are the a, b and ε.

  2. Later in Lemma 5.1.3: enter image description here

    here I don't understand how it magnifies the factor $(|\nabla\eta|+\eta)^2$ (it seems to suggest that $|\nabla\eta|+\eta\le 2^{(i-1)}$). Though I know $|\nabla\eta|\le2^iC$ and $0\le\eta\le1$, I need a further explanation.

Any hints are appreciated :)

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