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This extends a result of Dipierro-Farina-Valdinoci where the density estimates for such degenerate potentials were obtained for a bounded range of $m$'s. The original estimates for the classical case $p=m=2$ were established by"},"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":"2506.17000","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-06-20T13:51:55Z","cross_cats_sorted":[],"title_canon_sha256":"6c87eb49fa8a8a907009dccf8287297c4300b289655f7c099b01d71e2d5370d3","abstract_canon_sha256":"9cd0ff375a5a5382b3e70be0bae9f4170bc06fed2b449b368a192c0c7e9955f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:09:14.389492Z","signature_b64":"Yt9bxfbC4uP2cYTH11YvgHwxV0WRd03HKjPY+vbOwomHe8lPvlHlOsNgsfTIMY+aWoOWEN4Y+IkOORtmJcv1Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a8c98af42189f61ff1f0dc9c12fe6207727dafd5f89d64be514e49bce4c6e1a","last_reissued_at":"2026-06-11T01:09:14.388452Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:09:14.388452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Density estimates for Ginzburg-Landau energies with degenerate double-well potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chilin Zhang, Ovidiu Savin","submitted_at":"2025-06-20T13:51:55Z","abstract_excerpt":"We consider a class of Allen-Cahn equations associated with Ginzburg-Landau energies involving degenerate double-well potentials that vanish of order $m$ at the minima \\begin{equation}\n  J(v,\\Omega)=\\int_{\\Omega}\\Big\\{|\\nabla v|^{p}+(1-v^{2})^{m}\\Big\\}dx,\\quad 1<p<m, \\end{equation} and establish density estimates for the level sets of nontrivial minimizers $|v| \\leq 1$. This extends a result of Dipierro-Farina-Valdinoci where the density estimates for such degenerate potentials were obtained for a bounded range of $m$'s. 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