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I am woking on Evans PDE problem 5.10. #15: Fix $\alpha>0$ and let $U=B^0(0,1)\subset \mathbb{R}^n$. Show there exists a constant $C$ depending only on $n$ and $\alpha$ such that $ \int_U u^2 dx \le C\int_U|Du|^2 dx, $ provided that $u\in W^{1,1}(U)$ satisfies $ |\{x\in U\ |\ u(x)=0\}|>\alpha. $ I would appreciate your helping me with this problem.

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    I am working on this problem too and would like to see how to prove this. In 1 dimensional case, one can bound the left hand side by derivative and I naively hope this works in general dimension with appropriate modification. – 2012-11-14

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