{"paper":{"title":"On the numerical approximation of $p$-Biharmonic and $\\infty$-Biharmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.NA","authors_text":"Nikos Katzourakis (Reading, Tristan Pryer (Reading, UK)","submitted_at":"2017-01-25T18:20:29Z","abstract_excerpt":"In [KP16] (arXiv:1605.07880) the authors introduced a second-order variational problem in $L^{\\infty}$. The associated equation, coined the $\\infty$-Bilaplacian, is a \\emph{third order} fully nonlinear PDE given by $\\Delta^2_\\infty u\\, := (\\Delta u)^3 | D (\\Delta u) |^2 = 0.$ In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call $\\infty$-Biharmonic functions. For fixed $p$ we design a mixed finite element scheme for the pre-limiting equation, the $p$-Bilaplacian $\\Delta^2_p u\\, := \\Delta(| \\Delta u |^{p-2} \\Delta u) = 0.$ We prove c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07415","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":""},"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"}