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.
An application of Poincare inequality [solved]
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pde
functional-inequalities
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0Try Poincare's inequality. – 2012-11-12
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0I know it will be probably proved by Poincare inequality but don't know how to use the assumption on $u$. – 2012-11-12
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0I changed the title so that people can understand what I am looking for. – 2012-11-12
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0I misunderstood something and this can be easily solved. Thank you for your assistance. – 2012-11-13
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0Yeah. That happens! You are welcome. – 2012-11-13
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0I 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