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September 26, 2025

Density Estimates for a Nonlocal Variational Model with a Degenerate Double-Well Potential via the Sobolev Inequality

Key Points

Density estimates indicate the structure of level sets in minimizers influenced by a degenerate double-well potential.
New barriers facilitate the application of the fractional Sobolev inequality, enhancing the understanding of nonlocal variational models.
Robust proofs extend to cases involving quasilinear nonlocal equations driven by the fractional p-Laplacian, establishing foundational results.
Findings have implications for understanding phase separation behaviors in variational models with degenerate potentials.

Abstract

. We provide density estimates for level sets of minimizers of the energy \ (12 |u (x) -u (y) |ᵖ|x-y|^{n+sp}\, dx\, dy + ₑ䂞 |u (x) -u (y) |ᵖ|x-y|^{n+sp}\, dx\, dy+ W (u (x) ) \, dx\) where \ (p (1, +) \), \ (s (0, 1p) \) and \ (W\) is a double-well potential with polynomial growth \ (m [p, +) \) from the minima. These kinds of potentials are "degenerate", since they detach "slowly" from the minima, therefore they provide additional difficulties if one wishes to determine the relative sizes of the "layers" and the "pure phases". To overcome these challenges, we introduce new barriers allowing us to rely on the fractional Sobolev inequality and on a suitable iteration method. The proofs presented here are robust enough to consider the case of quasilinear nonlocal equations driven by the fractional \ (p\) -Laplacian, but our results are new even for the case \ (p=2\). Keywordsnonlocal energiesfractional Laplaciandegenerate potentialMSC codes47G1047B3435R1135B08

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