{"paper":{"title":"The weighted isoperimetric inequality and Sobolev inequality outside convex sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiali Lan, Lu Chen","submitted_at":"2025-10-02T04:05:32Z","abstract_excerpt":"In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $\\lambda_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $\\alpha$, such that $w^{1/\\alpha}$ is concave on $\\R^n$.\n  Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted P\\'{o}lya-Szeg\\\"{o} principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.01647","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.01647/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}