{"paper":{"title":"Fourier-based potential theory without an explicit Green's function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts.","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.AP","authors_text":"Fredrik Fryklund","submitted_at":"2026-04-13T13:20:09Z","abstract_excerpt":"Integral equation methods provide an effective framework for solving partial differential equations, but their applicability typically relies on the availability of explicit free-space Green's functions. For coupled systems arising in multiphysics applications, such Green's functions are generally not available, limiting the scope of classical potential theory-based approaches. In this work, we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regul"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The parabolic regularization produces a valid decomposition into nonlocal and localized components whose asymptotic expansions remain accurate for small ε, under the assumption that the operator belongs to the class of strongly elliptic systems.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A Fourier-symbol-based potential theory with parabolic regularization decomposes solutions and provides asymptotic expansions for volume, single-layer, and double-layer potentials without explicit Green's functions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e455fd53d59ab65552d61f2e38f7494dda6758bb8c815f6c251c0a109a9f3d70"},"source":{"id":"2604.11436","kind":"arxiv","version":2},"verdict":{"id":"4565baef-a8ea-4ffa-8e10-8f1b0f81b267","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:40:56.700720Z","strongest_claim":"we introduce a formulation of potential theory that avoids explicit use of Green's functions entirely, relying instead on the Fourier symbol of the governing operator. The central idea is a parabolic regularization of the symbol, which yields a decomposition of the solution into a smooth, nonlocal component and a spatially localized residual.","one_line_summary":"A Fourier-symbol-based potential theory with parabolic regularization decomposes solutions and provides asymptotic expansions for volume, single-layer, and double-layer potentials without explicit Green's functions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The parabolic regularization produces a valid decomposition into nonlocal and localized components whose asymptotic expansions remain accurate for small ε, under the assumption that the operator belongs to the class of strongly elliptic systems.","pith_extraction_headline":"Potential theory for elliptic PDEs can be formulated from the Fourier symbol alone by parabolic regularization that splits solutions into smooth nonlocal and localized parts."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11436/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"}