{"paper":{"title":"The regularity of a semilinear elliptic system with quadratic growth of gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ruiqi Jiang, Weiyong He","submitted_at":"2018-02-06T04:49:14Z","abstract_excerpt":"In this paper, we study semilinear elliptic systems with critical nonlinearity of the form \\begin{equation}\\label{sys01} \\Delta u=Q(x, u, \\nabla u), \\end{equation} for $u: \\mathbb{R}^n\\rightarrow \\mathbb{R}^K$, $Q$ has quadratic growth in $\\nabla u$. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When $n=2$, such a system does not have smooth regularity in general for $W^{1, 2}$ weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. H\\'elein (for $n=2$) and F. B\\'ethuel (for $n\\geq 3$), assert that a $W^{1, n}$ weak"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01796","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"}