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We set $K$, $\\Omega_+$ and $\\Omega_-$ to be respectively the zero-level set of $a$, {a>0} and {a<0}. Assuming $\\nabla a \\neq 0$ on $K$ and $a\\ne 0$ on $\\partial \\Omega$, we show that there exists a sequence $\\e_j \\to 0$ such that the above equation has a solution $u_{\\e_j}$ which converges uniformly t"},"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":"math/0702878","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2007-02-28T13:31:59Z","cross_cats_sorted":[],"title_canon_sha256":"ba3deb7cf1758b3249c8267fa596ac193937d4b644006fbb5b67fbb63646ac4f","abstract_canon_sha256":"e84f069861ca636624b580dfdc5e918f91f713ff8a46f3c454dd4052c419d4e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:23.285923Z","signature_b64":"OD4+gqcNyU6+/Q8u9vP0ksnBdB8aEvVc03hMLde9I9rawPqgfCXUUEWsWoGD9q04lP71tWBbavpaP9h36y7rAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89868bd97987ebc2f67c3ad59e06087f37766fced7fdb7f9ab8186f3b9f6331b","last_reissued_at":"2026-05-18T01:38:23.285203Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:23.285203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Transition Layer for the Heterogeneous Allen-Cahn Equation","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrea Malchiodi, Fethi Mahmoudi, Juncheng Wei","submitted_at":"2007-02-28T13:31:59Z","abstract_excerpt":"We consider the equation $\\e^{2}\\Delta u=(u-a(x))(u^2-1)$ in $\\Omega$, $\\frac{\\partial u}{\\partial \\nu} =0$ on $\\partial \\Omega$, where $\\Omega$ is a smooth and bounded domain in $\\R^n$, $\\nu$ the outer unit normal to $\\pa\\Omega$, and $a$ a smooth function satisfying $-1<a(x)<1$ in $\\ov{\\Omega}$. We set $K$, $\\Omega_+$ and $\\Omega_-$ to be respectively the zero-level set of $a$, {a>0} and {a<0}. Assuming $\\nabla a \\neq 0$ on $K$ and $a\\ne 0$ on $\\partial \\Omega$, we show that there exists a sequence $\\e_j \\to 0$ such that the above equation has a solution $u_{\\e_j}$ which converges uniformly t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702878","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0702878","created_at":"2026-05-18T01:38:23.285305+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0702878v1","created_at":"2026-05-18T01:38:23.285305+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0702878","created_at":"2026-05-18T01:38:23.285305+00:00"},{"alias_kind":"pith_short_12","alias_value":"RGDIXWLZQ7V4","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"RGDIXWLZQ7V4F5T4","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"RGDIXWLZ","created_at":"2026-05-18T12:25:56.245647+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4","json":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4.json","graph_json":"https://pith.science/api/pith-number/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/graph.json","events_json":"https://pith.science/api/pith-number/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/events.json","paper":"https://pith.science/paper/RGDIXWLZ"},"agent_actions":{"view_html":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4","download_json":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4.json","view_paper":"https://pith.science/paper/RGDIXWLZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0702878&json=true","fetch_graph":"https://pith.science/api/pith-number/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/graph.json","fetch_events":"https://pith.science/api/pith-number/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/action/storage_attestation","attest_author":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/action/author_attestation","sign_citation":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/action/citation_signature","submit_replication":"https://pith.science/pith/RGDIXWLZQ7V4F5T4HLKZ4BQIP4/action/replication_record"}},"created_at":"2026-05-18T01:38:23.285305+00:00","updated_at":"2026-05-18T01:38:23.285305+00:00"}