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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1108.0035","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-07-30T03:29:51Z","cross_cats_sorted":[],"title_canon_sha256":"13c4016d2e54ba54c4ad700abb71293f39f42ab3edfa4269040da81ef24680be","abstract_canon_sha256":"b2e428c015367bf4f3e73833a7adee6089f1f3d43f01a765d37a90c8f12944b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:31.801822Z","signature_b64":"rQjW7U6h+FrWErYEFyj7oakKso+GfxspGXVFBkgf07eAjnVIL6RZHStSVZd+gDuRp0l3Cspz0dv7ObpQizcdCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"487810a054b7730bc96ec389b33f31625efef7955124d87d596b40b413215d36","last_reissued_at":"2026-05-18T04:16:31.801449Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:31.801449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\xi\\in\\mathbb{R}^n$. 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