{"paper":{"title":"On solutions to a class of degenerate equations with the Grushin operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto Ferrero, Laura Abatangelo, Paolo Luzzini","submitted_at":"2024-10-16T14:55:22Z","abstract_excerpt":"The Grushin Laplacian $- \\Delta_\\alpha $ is a degenerate elliptic operator in $\\mathbb{R}^{h+k}$ that degenerates on $\\{0\\} \\times \\mathbb{R}^k$. We consider weak solutions of $- \\Delta_\\alpha u= Vu$ in an open bounded connected domain $\\Omega$ with $V \\in W^{1,\\sigma}(\\Omega)$ and $\\sigma > Q/2$, where $Q = h + (1+\\alpha)k$ is the so-called homogeneous dimension of $\\mathbb{R}^{h+k}$. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of $\\Omega$. As an application we derive strong unique continuation properties "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.12637","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.12637/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"}