engleski

Hausdorff dimension of singular sets of Sobolev functions and applications

We are interested in Sobolev functions with large singular sets in the sense of Hausdorff dimension. For functions in $W^{; ; ; k, p}; ; ; (R^N)$, $kp<N$, we find that the optimal bound is $N-kp$. Moreover, the bound is achieved, that is, there exists a class of maximally singular Sobolev functions. Similar results have been obtained for Besov spaces and Lizorkin-Triebel spaces. We describe a class of $p$-Laplace equations such that their weak solutions are singular on a prescribed fractal set having large Hausdorff dimension.

Sobolev function; singular sets; Hausdorff dimension

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