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.
Uniform estimates and limiting arguments for nonlocal minimal surfaces
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The fractional Massari functional Gamma-converges to the classical Massari functional, preserving minimizers, and yields limiting information for inhomogeneous Allen-Cahn equations together with the new notion of non-local hybrid mean curvature.
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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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$\Gamma$-convergence of the non-local Massari functional and applications to inhomogeneous Allen-Cahn equations
The fractional Massari functional Gamma-converges to the classical Massari functional, preserving minimizers, and yields limiting information for inhomogeneous Allen-Cahn equations together with the new notion of non-local hybrid mean curvature.