{"paper":{"title":"Min-max $n$-harmonic maps of degree 1 with free-boundary into $\\mathbb{S}^{n-1}$ in almost round balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dorian Martino, Katarzyna Mazowiecka, R\\'emy Rodiac","submitted_at":"2026-05-27T16:01:15Z","abstract_excerpt":"Let $n\\geq 3$ and let $\\Omega \\subset \\mathbb{R}^n$ be a $\\mathcal{C}^1$ bounded domain which is diffeomorphic to a ball. We investigate here the problem of finding critical points of the $n$-energy in the space $\\mathcal{I}=\\{v\\in W^{1,n}(\\Omega,\\mathbb{R}^n) ; \\ |\\mathrm{tr}_{|\\partial \\Omega}v|=1\\}$. Maps in $\\mathcal{I}$ have a well-defined topological degree on $\\partial \\Omega$ but this degree is not continuous for the weak convergence in $W^{1,n}$. Hence finding critical points with prescribed degrees results in a problem of lack of compactness. We first prove that minimizers of the $n$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28668","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/2605.28668/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"}