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I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this compact domain a "domain with Lipschitz boundary" ? I don't know any book I could look for the answer. Thanks

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  • $\begingroup$ Have you tried this one?Dacorogna, B. (2004). Introduction to the Calculus of Variations. Imperial College Press, London. ISBN 1-86094-508-2. $\endgroup$ Dec 17, 2013 at 0:11
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    $\begingroup$ I do not quit understand your question. What happens just take a domain D in the plane with a single cusp? The boundary of D is a rectifiable Jordan curve, but it is not Lipshichtz. $\endgroup$ Dec 17, 2013 at 7:37

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No. Take a spiral of finite length but spiraling infinitely many times around its endpoint. It is rectifiable but not Lipschitz. You can have infinitely many points on the boundary where the boundary behaves like that.

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