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More explicitly, if $\\epsilon\\ge0$ and $\\bar u_\\epsilon$ minimizes the functional $$ \\int_\\Omega(\\epsilon+|\\nabla_{\\H^n}u|^2)^{p/2}$$ among the functions with prescribed Dirichlet boundary condition that stay below a smooth obstacle $\\psi$, then \n0 \\le \\div_{\\H^n}\\, \\Big((\\epsilon+|\\nabla_{\\H^n}\\bar u_\\epsilon|^2)^{(p/2)-1} \\nabla_{\\H^n}\\bar u_\\epsilon\\Big) \n\\qquad \\le (\\div_{\\H^n}\\, \\Big((\\epsilon+|\\nabla_{\\H^n}\\psi|^2)^{(p/2)-1} \\nab"},"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":"1105.5075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-05-25T16:09:52Z","cross_cats_sorted":[],"title_canon_sha256":"a188306f1585e177bef07a86db3b3c1f6f4c82fe72efd9e2f95792ea02917d55","abstract_canon_sha256":"29b85e5a970dcee0cec15fc429fb60072afe1a06f95700fc60e266bdaca25b8b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:20.839504Z","signature_b64":"GdVz5MFZs0C7oFShytUDtagFg9wBYcrp7SBpi+yQD03WkHs2+t31MoXafWmIqt5s/mPbhDWJMJr/3hv1VpuJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b19c21544183f916cb5d796d8aae0ba85b9b36c2db1ca5c6f9168f2d65bbd7a","last_reissued_at":"2026-05-18T04:21:20.839070Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:20.839070Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Lewy-Stampacchia Estimate for quasilinear variational inequalities in the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrea Pinamonti, Enrico Valdinoci","submitted_at":"2011-05-25T16:09:52Z","abstract_excerpt":"We consider an obstacle problem in the Heisenberg group framework, and we prove that the operator on the obstacle bounds pointwise the operator on the solution. 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