{"paper":{"title":"Starshape of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Paolo Salani, Sven Jarohs, Tadeusz Kulczycki","submitted_at":"2017-01-09T21:53:57Z","abstract_excerpt":"In the present work we study solutions of the problem $-(-\\Delta)^{\\alpha/2}u = f(x,u)$ in $D_0\\setminus \\overline{D}_1$, with exterior conditions $u = 0$ in $R^N \\setminus D_0$ and $u = 1$ in $\\overline{D}_1$, where $D_1, D_0 \\subset R^N$ are open sets such that $\\overline{D}_1 \\subset D_0$, $\\alpha \\in (0,2)$, and $f$ is a nonlinearity. Under different assumptions on $f$ we prove that, if $D_0$ and $D_1$ are starshaped with respect to the same point $\\bar{x} \\in \\overline{D}_1$, then the same occurs for every superlevel set of $u$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02367","kind":"arxiv","version":2},"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"}