{"paper":{"title":"On the Schiffer and Berenstein conjectures with high-frequency for convex domains in the plane","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guowei Dai, Juncheng Wei, Yingxin Sun, Yong Zhang","submitted_at":"2025-11-25T01:16:08Z","abstract_excerpt":"In this paper, by introducing two-point stationary-phase amplitude defect, we provide a partial positive answer to the Schiffer and Berenstein conjectures in $\\mathbb{R}^2$. More precisely, assuming that a bounded uniformly convex domain $\\Omega \\subset \\mathbb{R}^2$ has a connected boundary of class $C^{2,\\epsilon}$ with $\\epsilon \\in (0,1)$, we show that if, for some nonzero constant $c_D$, the overdetermined elliptic problem \\begin{equation} -\\Delta u = \\alpha u \\ \\text{ in } \\ \\Omega, \\qquad u = 0 \\ \\text{ on } \\ \\partial\\Omega, \\qquad \\frac{\\partial u}{\\partial \\nu} = c_{D} \\ \\text{ on } "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.19819","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2511.19819/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}