{"paper":{"title":"Improvement of flatness for nonlocal phase transitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrico Valdinoci, Joaquim Serra, Serena Dipierro","submitted_at":"2016-11-30T11:52:05Z","abstract_excerpt":"We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations that includes $(-\\Delta)^{s/2} u = u-u^3$, with~$s\\in(0,1)$, we obtain a result in the same spirit of a celebrated theorem of Savin for the equation $-\\Delta u = u-u^3$. As a consequence, we deduce that entire solutions to~$(-\\Delta)^{s/2} u = u-u^3$ with asymptotically flat level sets are $1$D when~$s\\in(0,1)$.\n  The results presented are completely new even for the case of the fractional Laplacian, but the robustness of the proofs allows us to treat also more general, possibly anisotr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.10105","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}