Nonlocal minimal surfaces are unique for all but countably many parameters in any strictly increasing family of exterior data, and arbitrarily small perturbations yield smooth minimizers in one dimension beyond the critical threshold.
Pure Appl
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Viscosity subsolutions to nonlocal mean curvature-type equations satisfy universal volumetric estimates at all scales, and low-density ones necessarily have topological boundaries with positive Lebesgue measure.
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Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension
Nonlocal minimal surfaces are unique for all but countably many parameters in any strictly increasing family of exterior data, and arbitrarily small perturbations yield smooth minimizers in one dimension beyond the critical threshold.
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Volumetric density estimates for nonlocal minimal surfaces
Viscosity subsolutions to nonlocal mean curvature-type equations satisfy universal volumetric estimates at all scales, and low-density ones necessarily have topological boundaries with positive Lebesgue measure.