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What can we say about the boundary of the level set of a Sobolev function?
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I figured out the problem later and until today I could have time to write it down. The proof of the rectifiability of free boundary $\partial \{u>0\}$ requires the Lipchitz regularity of $u$ ...

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minimizing weighted length of closed curves
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I just found out two proofs. One is analytic, and the other is geometric. For the analytic proof, one can use polar coordinates and apply some elemetary inequality. But such proof has limited ...

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