{"paper":{"title":"Compactness properties and ground states for the affine Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Cyril Tintarev, Ian Schindler","submitted_at":"2017-08-08T09:09:19Z","abstract_excerpt":"The paper studies compactness properties of the affine Sobolev inequality of Gaoyong Zhang et al in the case $p=2$, and existence and regularity of related minimizers, in particular, solutions to the nonlocal Dirichlet problems \\[ -\\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\\frac{\\partial^2u}{\\partial x_i\\partial x_j}=f \\mbox{ in }\\Omega\\subset\\mathbb R^N, \\] and \\[ -\\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\\frac{\\partial^2u}{\\partial x_i\\partial x_j}=u^{q-1}\\,,\\quad u>0,\\mbox{ in }\\Omega\\subset\\mathbb R^N, \\] where $A_{ij}[u]=\\int_\\Omega\\frac{\\partial u}{\\partial x_i}\\frac{\\partial u}{\\partial x_j}\\mathrm{d}x$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02413","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"}