This analysis uncovers Sobolev and Poincaré inequalities in non-local Sobolev functions, suggesting insights into isoperimetric problems.
We investigate the space of non-local Sobolev functions associated with an integral kernel. We prove an extension result, Sobolev and Poincaré inequalities and an isoperimetric inequality for the non-local perimeter restricted to a set. Finally, we remark on non-local isoperimetric problems, even when the underlying kernel is not necessarily radially symmetric.
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Bessas et al. (2025) studied this question.
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