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In this paper, by generalizing Ben Andrews's result we obtain the complete classification of the solutions of the planar isotropic $L_p$ dual Minkowski problem, that is, for any $p,q\\in\\mathbb{R}$ we obtain the complete classification of the solutions of the following equation: \\begin{equation*}\n  u^{1-p}(u_{\\theta}^2+u^2)^{\\frac{q-2}{2}}(u_{\\theta\\theta}+u)=1\\quad\\text{on}\\ \\mathbb{S}^1. \\end{equation*} To establish the classification, we convert the O"},"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":"2209.14630","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-09-29T08:46:01Z","cross_cats_sorted":["math.AP","math.MG"],"title_canon_sha256":"d9392c840bca941ea29aa1eac6e559b820710473ae7e45160a0a8c749a221c09","abstract_canon_sha256":"55b516bf89d1753958369f0d795b97a0341939f2d8009e089107e50d4423d5c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:02:14.394131Z","signature_b64":"qznRAH2/coLyCjbY3HoRXjwd72PlBuoI51vxLF8SE8wwXsuJY8AKcfamlmQag7SGNGQHxsJxgo8nfLvTePNpAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05b102ab02db1543d02bd66936775314fdd5c0c445436e9ada3291ffe876fd17","last_reissued_at":"2026-07-05T05:02:14.392508Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:02:14.392508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classification of solutions for the planar isotropic $L_p$ dual Minkowski problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP","math.MG"],"primary_cat":"math.DG","authors_text":"Haizhong Li, Yao Wan","submitted_at":"2022-09-29T08:46:01Z","abstract_excerpt":"In his beautiful paper [1], Ben Andrews obtained the complete classification of the solutions of the planar isotropic $L_p$ Minkowski problem. 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