A Geometric B-Spline Approach to Mountain-Pass Type Solutions of Nonlinear Dirichlet Problems

A Geometric B-Spline Approach to Mountain-Pass Type Solutions of Nonlinear Dirichlet Problems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Geometric B-Spline Approach to Mountain-Pass Type Solutions of Nonlinear Dirichlet Problems Boróka Olteán-Péter This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8617563/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract We study the numerical computation of nontrivial (typically saddle-type) critical points of variational functionals associated with nonlinear Dirichlet problems involving the $p$-Laplacian. While classical numerical mountain pass algorithms rely on path deformations or finite element discretizations, we propose a different approach based on a geometric B-spline representation of the solution. The idea is to parameterize the function space by smooth spline curves and perform a mountain-pass type \emph{up--down} iteration directly in the space of control points. This transforms the infinite-dimensional mountain pass problem into a finite-dimensional geometric problem and introduces tools from spline theory into nonlinear elliptic PDE computation. The descent direction is obtained through an auxiliary Poisson equation, yielding a Sobolev gradient that significantly stabilizes the iteration. Convergence is monitored via both the gradient norm and the Euler--Lagrange residual, ensuring that the resulting B-spline approximation satisfies the underlying PDE. Numerical experiments for the model case p = 2 and f(u) = u 3 on Ω =(0, 1) demonstrate that the method computes nontrivial solutions consistent with mountain-pass geometry; depending on initialization, the computed profiles may include sign-changing solutions. The results highlight the effectiveness and flexibility of B-spline geometry in nonlinear variational problems and suggest new directions for numerical mountain pass methods based on geometric parametrizations. B-Spline Optimization Problem Mountain Pass Theorem Dirichlet Problem Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 02 Mar, 2026 Reviews received at journal 25 Feb, 2026 Reviews received at journal 26 Jan, 2026 Reviewers agreed at journal 22 Jan, 2026 Reviewers agreed at journal 20 Jan, 2026 Reviewers invited by journal 18 Jan, 2026 Editor assigned by journal 16 Jan, 2026 Submission checks completed at journal 16 Jan, 2026 First submitted to journal 16 Jan, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Problems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"acta-universitatis-sapientiae-informatica","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Acta Universitatis Sapientiae, Informatica](https://link.springer.com/journal/44427)","snPcode":"44427","submissionUrl":"https://submission.springernature.com/new-submission/44427/3","title":"Acta Universitatis Sapientiae, Informatica","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"B-Spline, Optimization Problem, Mountain Pass Theorem, Dirichlet Problem","lastPublishedDoi":"10.21203/rs.3.rs-8617563/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8617563/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe study the numerical computation of nontrivial (typically saddle-type) critical points of variational functionals associated with nonlinear Dirichlet problems involving the $p$-Laplacian. While classical numerical mountain pass algorithms rely on path deformations or finite element discretizations, we propose a different approach based on a geometric B-spline representation of the solution. The idea is to parameterize the function space by smooth spline curves and perform a mountain-pass type \\emph{up--down} iteration directly in the space of control points. This transforms the infinite-dimensional mountain pass problem into a finite-dimensional geometric problem and introduces tools from spline theory into nonlinear elliptic PDE computation. The descent direction is obtained through an auxiliary Poisson equation, yielding a Sobolev gradient that significantly stabilizes the iteration. Convergence is monitored via both the gradient norm and the Euler--Lagrange residual, ensuring that the resulting B-spline approximation satisfies the underlying PDE. Numerical experiments for the model case p = 2 and f(u) = u\u003csup\u003e3\u003c/sup\u003e on Ω =(0, 1) demonstrate that the method computes nontrivial solutions consistent with mountain-pass geometry; depending on initialization, the computed profiles may include sign-changing solutions. The results highlight the effectiveness and flexibility of B-spline geometry in nonlinear variational problems and suggest new directions for numerical mountain pass methods based on geometric parametrizations.\u003c/p\u003e","manuscriptTitle":"A Geometric B-Spline Approach to Mountain-Pass Type Solutions of Nonlinear Dirichlet Problems","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-20 13:06:34","doi":"10.21203/rs.3.rs-8617563/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-03-02T20:12:01+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-25T11:07:33+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-01-26T15:13:39+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"11249860044111039887142763500914657250","date":"2026-01-22T18:22:26+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"245776472418156577744043816809864691465","date":"2026-01-20T13:47:59+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-01-19T00:59:43+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-01-16T16:26:25+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-01-16T16:25:15+00:00","index":"","fulltext":""},{"type":"submitted","content":"Acta Universitatis Sapientiae, Informatica","date":"2026-01-16T09:38:32+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"acta-universitatis-sapientiae-informatica","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Acta Universitatis Sapientiae, Informatica](https://link.springer.com/journal/44427)","snPcode":"44427","submissionUrl":"https://submission.springernature.com/new-submission/44427/3","title":"Acta Universitatis Sapientiae, Informatica","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"1fc4f052-beba-4675-8666-39972b2bf10c","owner":[],"postedDate":"January 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-03-19T02:27:53+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-20 13:06:34","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8617563","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8617563","identity":"rs-8617563","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}