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In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution. In 2015, when the second edition of the Scottish Book with updates and commentary on ...

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Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by \begin{align*} \text{Cap}_{p}(K, U) := \inf \left\{ \int_U |\...