{"paper":{"title":"Some Energy Estimates for Stable Solutions to Fractional Allen-Cahn Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changfeng Gui, Qinfeng Li","submitted_at":"2019-04-16T03:54:11Z","abstract_excerpt":"In this paper we study stable solutions to the fractional equation \\begin{align}\n  (-\\Delta)^s u =f(u), \\quad |u| < 1 \\quad \\mbox{in $\\mathbb{R}^d$}, \\end{align}where $0<s<1$ and $f:[-1,1] \\rightarrow \\mathbb{R}$ is a $C^{1,\\alpha}$ function for $\\alpha>\\max\\{0, 1-2s\\}$. We obtain sharp energy estimates for $0<s<1/2$ and rough energy estimates for $1/2 \\le s <1$. These lead to a different proof from literature of the fact that when $d=2, \\, 0<s<1$, entire stable solutions are $1$-D solutions.\n  The scheme used in this paper is inspired by Cinti-Serra-Valdinoci[CSV17] which deals with stable no"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.07443","kind":"arxiv","version":1},"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"}