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I'm studying Elliptic Partial Differential Equations by Q. Han and F. Lin. In Lemma 1.41 is given the elliptic equation $D_j(a_{ij}D_i u)=0$ where the coefficient matrix $(a_{ij})$ is constant positive definite and elliptic (with the bound $0<\lambda\leq\Lambda$). There is an energy estimate of its weak solution $u\in C^1(B_1)$ which reads $$\int_{B_\rho}u^2\leq c\rho^n\int_{B_1}u^2$$ where $c=c(\lambda,\Lambda)$. Is there any reason that the case $1/2<\rho\leq 1$ is trivial? I can hardly find out why it is evident (to the authors). If so, why does this inequality holds in this case?

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  • $\begingroup$ Does the maximum principle hold for this PDE? $\endgroup$ – user7530 Aug 17 '13 at 15:19
  • $\begingroup$ Yes, it does. Does this fact explain anything? $\endgroup$ – Sunghan Kim Aug 17 '13 at 16:33
  • $\begingroup$ I just added the reference in the content. Thanks for your advice. $\endgroup$ – Sunghan Kim Aug 18 '13 at 0:44
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The authors neglect to mention that $c$ depends on dimension $n$ as well. It should have been $c=c(\lambda,\Lambda,n)$. Indeed, if you look at the proof on page 23, you'll see that

  • in Method 1, they use a linear change of variables. This contributes the Jacobian determinant, which depends on $n$.
  • in Method 2, they choose $k$ depending on $n$, and use a constant that depends on $k$

Now that the dependency of $c$ on $n$ is admitted, we can take $c\ge 2^n$ to satisfy the inequality for $\rho\in [1/2,1]$.

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    $\begingroup$ I was also thinking that $c$ must depend on $n$ as well as $\lambda$ and $\Lambda$. Moreover, the inequality (in Method 1) $$\int_{B_\rho}|u|^2\leq \rho^n\|u\|_{L^\infty(B_{1/2})}$$ is also incorrect; it is a H\"{o}lder inequality so we need to multiply the right side by $\omega_n=|B_1|$. Thanks for your help. $\endgroup$ – Sunghan Kim Aug 19 '13 at 23:58

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