A counterexample regarding a two-phase problem for harmonic measure in VMO

A counterexample regarding a two-phase problem for harmonic measure in VMO | 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A counterexample regarding a two-phase problem for harmonic measure in VMO Xavier Tolsa This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4095167/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 May, 2025 Read the published version in Potential Analysis → Version 1 posted 7 You are reading this latest preprint version Abstract Let $\Omega^+\subset\mathbb R^{n+1}$ be a vanishing Reifenberg flat domain such that $\Omega^+$ and $\Omega^-=\mathbb R^{n+1}\setminus\overline {\Omega^+}$ have joint big pieces of chord-arc subdomains and such that the outer unit normal to $\partial\Omega^+$ belongs to $VMO(\omega^+)$, where $\omega^\pm$ is the harmonic measure in $\Omega^\pm$. Up to now it was an open question if these conditions imply that $\log\dfrac{d\omega^-}{d\omega^+} \in VMO(\omega^+)$. In this paper we answer this question in the negative by constructing an appropriate counterexample in $\mathbb R^2$, with the additional property that the outer unit normal to $\partial\Omega^+$ is constant $\omega^+$-a.e.\ in $\partial\Omega^+$. harmonic measure two-phase problem Reifenberg flat domains Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 03 May, 2025 Read the published version in Potential Analysis → Version 1 posted Editorial decision: Revision requested 10 Aug, 2024 Reviews received at journal 17 Jul, 2024 Reviewers agreed at journal 27 Mar, 2024 Reviewers invited by journal 27 Mar, 2024 Editor assigned by journal 27 Mar, 2024 Submission checks completed at journal 25 Mar, 2024 First submitted to journal 13 Mar, 2024 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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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4095167","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":283411058,"identity":"b0783aad-3325-4dc7-825f-ad30d86ce79a","order_by":0,"name":"Xavier Tolsa","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA40lEQVRIie3PMQrCMACF4VcKZglkTfESFUEQ1LNUCp2qIIKLoIVCXXqAegsP4BDJ0EV0dVQKulZcBB0s1cEpdnTID8mUj0cAne5PE+VNipMDTmXCYQJG8iZGUIWVxKRViJ1uG+K6xoyZ5HTrRr0hiLwcVcSKfXuTnMGtkDbrg8gdg3oN5QqD50gqwG1JURCzH3CoSY2dHfksCcke7WheEHJVr3BXSJQErboRyYJQ9YqVZGITC24tQ9pqx7t0XKP+KFERe98P87voMEYW2eE+mQ4ZSVe5inziX7+r8Fyn0+l0P3oBQg1F+mP2ozkAAAAASUVORK5CYII=","orcid":"","institution":"Institució Catalana de Recerca i Estudis Avançats","correspondingAuthor":true,"prefix":"","firstName":"Xavier","middleName":"","lastName":"Tolsa","suffix":""}],"badges":[],"createdAt":"2024-03-13 17:59:24","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4095167/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4095167/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11118-025-10214-3","type":"published","date":"2025-05-03T15:57:47+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":81987802,"identity":"72a8995c-7dde-4588-8e48-44eb2d4076b9","added_by":"auto","created_at":"2025-05-05 16:06:06","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":332745,"visible":true,"origin":"","legend":"","description":"","filename":"contravmo4.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4095167/v1_covered_c9fbe581-8bf4-4023-b7f2-810836fbd305.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A counterexample regarding a two-phase problem for harmonic measure in VMO","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":"potential-analysis","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"pota","sideBox":"Learn more about [Potential Analysis](http://link.springer.com/journal/11118)","snPcode":"11118","submissionUrl":"https://submission.nature.com/new-submission/11118/3","title":"Potential Analysis","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"harmonic measure, two-phase problem, Reifenberg flat domains","lastPublishedDoi":"10.21203/rs.3.rs-4095167/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4095167/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Let $\\Omega^+\\subset\\mathbb R^{n+1}$ be a vanishing Reifenberg flat domain such that $\\Omega^+$ and $\\Omega^-=\\mathbb R^{n+1}\\setminus\\overline {\\Omega^+}$ have joint big pieces of chord-arc subdomains and such that the outer unit normal to $\\partial\\Omega^+$ belongs to $VMO(\\omega^+)$, where $\\omega^\\pm$ is the harmonic measure in $\\Omega^\\pm$. 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