{"paper":{"title":"Rigidity of pairs of quasiregular mappings whose symmetric part of gradient are close","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrew Lorent","submitted_at":"2013-12-02T06:35:48Z","abstract_excerpt":"For $A\\in M^{2\\times 2}$ let $S(A)=\\sqrt{A^T A}$, i.e. the symmetric part of the polar decomposition of $A$. We consider the relation between two quasiregular mappings whose symmetric part of gradient are close. Our main result is the following. Suppose $v,u\\in W^{1,2}(B_1(0):\\mathbb{R}^2)$ are $Q$-quasiregular mappings with $\\int_{B_1(0)} \\det(Du)^{-p} dz\\leq C_p$ for some $p\\in (0,1)$ and $\\int_{B_1(0)} |Du|^2 dz\\leq 1$. There exists constant $M>1$ such that if $$ \\int_{B_1(0)} |S(Du)-S(Dv)|^2 dz=\\epsilon $$ then $$ \\int_{B_{\\frac{1}{2}}(0)} |Dv-R Du| dz\\leq c C_p^{\\frac{1}{p}}\\epsilon^{\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0339","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"}