{"paper":{"title":"Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Scott Rodney","submitted_at":"2011-07-30T03:29:51Z","abstract_excerpt":"This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form {eqnarray} \\nabla'P(x)\\nabla u +{\\bf HR}u+{\\bf S'G}u +Fu &=& f+{\\bf T'g} \\textrm{in}\\Theta u&=&\\phi\\textrm{on}\\partial \\Theta.{eqnarray} The principal part $\\xi'P(x)\\xi$ of the above equation is assumed to be comparable to a quadratic form ${\\cal Q}(x,\\xi) = \\xi'Q(x)\\xi$ that may vanish for non-zero $\\xi\\in\\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\\Theta)=W^{1,2}(\\Omega,Q)$ and $QH^1_0(\\Theta)=W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0035","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"}