{"paper":{"title":"Symmetry via antisymmetric maximum principles in nonlocal problems of variable order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Sven Jarohs, Tobias Weth","submitted_at":"2014-06-24T09:48:42Z","abstract_excerpt":"We consider the nonlinear problem \\[(P) \\;\\; I u=f(x,u) \\text{ in $\\Omega$,} \\;\\; u=0 \\text{ on $\\mathbb{R}^{N}\\setminus\\Omega$ }\\] in an open bounded set $\\Omega\\subset\\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may be anisotropic and may have varying order. We assume mild symmetry and monotonicity assumptions on $I$, $\\Omega$ and the nonlinearity $f$ with respect to a fixed direction, say $x_1$, and we show that any nonnegative weak solution $u$ of $(P)$ is symmetric in $x_1$. Moreover, we have the following alternative: Either $u\\equiv 0$ in $\\Omega$, or $u$ is strictly decreasi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6181","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"}