Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields

Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields | 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 Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields Natalia Zorii This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2433330/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 22 Jul, 2023 Read the published version in Potential Analysis → Version 1 posted 7 You are reading this latest preprint version Abstract Note: Please see pdf for full abstract with equations. For the Riesz kernel κ α (x, y) := |x − y| α−n of order 0 < α < n on R n , n ⩾ 2 , we introduce the so-called inner pseudo-balayage ˆω A of a (Radon) measure ω on R n to a set A ⊂ R n as the (unique) measure minimizing the Gauss functional ∫ κ α (x, y) d(μ ⊗ μ)(x, y) − 2 ∫ κ α (x, y) d(ω ⊗ μ)(x, y) over the class ε + (A) of all positive measures μ of finite energy, concentrated on A . For quite general signed ω (not necessarily of finite energy) and A (not necessarily closed), such ˆω A does exist, and it maintains the basic features of inner balayage for positive measures (defined when α ⩽ 2 ), except for those implied by the domination principle. (To illustrate the latter, we point out that, in contrast to what occurs for the balayage, the inner pseudo-balayage of a positive measure may increase its total mass.) The inner pseudo-balayage ˆω A is further shown to be a powerful tool in the problem of minimizing the Gauss functional over all μ ∈ ε + (A) with μ(R n ) = 1 , which enables us to improve substantially many recent results on this topic, by strengthening their formulations and/or by extending the areas of their applications. For instance, if A is a quasiclosed set of nonzero inner capacity c ∗ (A) , and if ω is a signed measure, compactly supported in R n \ Cl Rn A , then the problem in question is solvable if and only if either c ∗ (A) < ∞, or ˆω A (R n ) ⩾ 1 . In particular, if c ∗ (A) = ∞ , then the problem has no solution whenever ω+(R n ) < 1/C n,α , where C n,α := 1 if α ⩽ 2 , and C n,α := 2 n−α otherwise; whereas ω − (R n ) , the total amount of the negative charge, has no influence on this phenomenon. The results obtained are illustrated by some examples. 2010 Mathematics Subject Classification: Primary 31C15. Minimum Riesz energy problems with external fields inner Riesz balayage inner Riesz pseudo-balayage. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 22 Jul, 2023 Read the published version in Potential Analysis → Version 1 posted Editorial decision: Major revision 14 May, 2023 Reviews received at journal 10 May, 2023 Reviewers agreed at journal 12 Feb, 2023 Reviewers invited by journal 08 Feb, 2023 Editor assigned by journal 12 Jan, 2023 Submission checks completed at journal 02 Jan, 2023 First submitted to journal 01 Jan, 2023 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. 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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-2433330","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":164447770,"identity":"c65e6754-1880-4d27-be10-1eab93b199da","order_by":0,"name":"Natalia Zorii","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8ElEQVRIiWNgGAWjYLCCBwYQ+kBCBZBkZm4goJ6ZgSEBruUMSICRGC0wNmMbmMSvxZz9/MEPCQV2DPLtvQ8PPJxXG83fDtTyo2IbTi2WPcnMEgkGyQyMPccNDiRuO5474zBjA2PPmds4tRgcSGYAamFmYJZIYwBqOZbbANTCzNiGR8v5x8w/EgzqGdjAWuYcy51PUMuNZDagLYcZeMBaGmpyNxDW8tjMIsHgOI8EzzFgIB87kLsRqOUgXr+cT3x848Ofajn59jbmjz9q6nLnnT988MGPCtxaYIAHSh8GkwcIqkcCdaQoHgWjYBSMghECAKtVWjN/D1rLAAAAAElFTkSuQmCC","orcid":"","institution":"Institute of Mathematics, National Academy of Sciences of Ukraine","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Natalia","middleName":"","lastName":"Zorii","suffix":""}],"badges":[],"createdAt":"2023-01-01 12:59:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2433330/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2433330/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11118-023-10087-4","type":"published","date":"2023-07-22T21:41:11+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":44733534,"identity":"97fbb257-22ab-463a-a404-4f9c7d1b7366","added_by":"auto","created_at":"2023-10-16 22:07:45","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":708001,"visible":true,"origin":"","legend":"","description":"","filename":"Pseudobalayage.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2433330/v1_covered_74c5d350-266f-4c63-ab6b-291902796494.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields","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":"Minimum Riesz energy problems with external fields, inner Riesz balayage, inner Riesz pseudo-balayage.","lastPublishedDoi":"10.21203/rs.3.rs-2433330/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2433330/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eNote: Please see pdf for full abstract with equations.\u003c/p\u003e\n\u003cp\u003eFor the Riesz kernel \u003cem\u003eκ\u003c/em\u003e\u003csub\u003e\u003cem\u003eα\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e(x, y) := |x − y|\u003c/em\u003e\u003csup\u003e\u003cem\u003eα−n\u003c/em\u003e\u003c/sup\u003e of order \u003cem\u003e0 \u0026lt; α \u0026lt; n\u003c/em\u003e on \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e, n ⩾ 2\u003c/em\u003e, we introduce the so-called inner pseudo-balayage \u003cem\u003eˆω\u003c/em\u003e\u003csup\u003e\u003cem\u003eA\u003c/em\u003e\u003c/sup\u003e of a (Radon) measure \u003cem\u003eω \u003c/em\u003eon \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e \u003c/em\u003eto a set \u003cem\u003eA ⊂ R\u003c/em\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\u0026nbsp;as the (unique) measure minimizing the Gauss functional\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e∫ κ\u003c/em\u003e\u003csub\u003e\u003cem\u003eα\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e(x, y) d(μ ⊗ μ)(x, y) − 2 ∫ κ\u003c/em\u003e\u003csub\u003e\u003cem\u003eα\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e(x, y) d(ω ⊗ μ)(x, y)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eover the class \u003cem\u003eε\u003c/em\u003e\u003csup\u003e\u003cem\u003e+\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e(A)\u003c/em\u003e of all positive measures \u003cem\u003eμ \u003c/em\u003eof finite energy, concentrated on \u003cem\u003eA\u003c/em\u003e. For quite general signed \u003cem\u003eω \u003c/em\u003e(not necessarily of finite energy) and \u003cem\u003eA \u003c/em\u003e(not necessarily closed), such \u003cem\u003eˆω\u003c/em\u003e\u003csup\u003e\u003cem\u003eA\u003c/em\u003e\u003c/sup\u003e does\u0026nbsp;exist, and it maintains the basic features of inner balayage for positive measures (defined when\u0026nbsp;\u003cem\u003eα ⩽ 2\u003c/em\u003e), except for those implied by the domination principle. (To illustrate the latter, we point out\u0026nbsp;that, in contrast to what occurs for the balayage, the inner pseudo-balayage of a positive measure\u0026nbsp;may increase its total mass.) The inner pseudo-balayage \u003cem\u003eˆω\u003c/em\u003e\u003csup\u003e\u003cem\u003eA\u003c/em\u003e\u003c/sup\u003e is further shown to be a powerful tool in\u0026nbsp;the problem of minimizing the Gauss functional over all \u003cem\u003eμ ∈ ε\u003c/em\u003e\u003csup\u003e\u003cem\u003e+\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e(A)\u003c/em\u003e with \u003cem\u003eμ(R\u003c/em\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e) = 1\u003c/em\u003e, which enables\u0026nbsp;us to improve substantially many recent results on this topic, by strengthening their formulations\u0026nbsp;and/or by extending the areas of their applications. For instance, if A is a quasiclosed set of nonzero\u0026nbsp;inner capacity \u003cem\u003ec\u003c/em\u003e\u003csub\u003e\u003cem\u003e∗\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e(A)\u003c/em\u003e, and if \u003cem\u003eω \u003c/em\u003eis a signed measure, compactly supported in \u003cem\u003eR\u003c/em\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e \\ Cl\u003c/em\u003e\u003csub\u003e\u003cem\u003eRn\u003c/em\u003e\u003c/sub\u003e\u003cem\u003eA\u003c/em\u003e, then the\u0026nbsp;problem in question is solvable if and only if either \u003cem\u003ec\u003c/em\u003e\u003csub\u003e\u003cem\u003e∗\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e(A) \u0026lt; ∞, \u003c/em\u003eor \u003cem\u003eˆω\u003c/em\u003e\u003csup\u003e\u003cem\u003eA\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e(R\u003c/em\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e) ⩾ 1\u003c/em\u003e. In particular,\u0026nbsp;if \u003cem\u003ec\u003c/em\u003e\u003csub\u003e\u003cem\u003e∗\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e(A) = ∞\u003c/em\u003e, then the problem has no solution whenever \u003cem\u003eω+(R\u003c/em\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e) \u0026lt; 1/C\u003c/em\u003e\u003csub\u003e\u003cem\u003en,α\u003c/em\u003e\u003c/sub\u003e, where \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003en,α\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e := 1\u003c/em\u003e if\u0026nbsp;\u003cem\u003eα ⩽ 2\u003c/em\u003e, and \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003en,α\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e := 2\u003c/em\u003e\u003csup\u003e\u003cem\u003en−α\u003c/em\u003e\u003c/sup\u003e otherwise; whereas \u003cem\u003eω\u003c/em\u003e\u003csup\u003e\u003cem\u003e−\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e(R\u003c/em\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e)\u003c/em\u003e, the total amount of the negative charge, has\u0026nbsp;no influence on this phenomenon. The results obtained are illustrated by some examples.\u003c/p\u003e\n\u003cp\u003e2010 Mathematics Subject Classification: Primary 31C15.\u003c/p\u003e","manuscriptTitle":"Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-01-05 15:23:47","doi":"10.21203/rs.3.rs-2433330/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major revision","date":"2023-05-15T00:59:24+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2023-05-10T16:33:37+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"b789b838-667b-4823-b3f7-c886b8a26d6f","date":"2023-02-12T10:38:11+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2023-02-09T01:41:39+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2023-01-13T01:34:08+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2023-01-03T02:23:35+00:00","index":"","fulltext":""},{"type":"submitted","content":"Potential Analysis","date":"2023-01-01T12:52:57+00:00","index":"","fulltext":""}],"status":"published","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}}],"origin":"","ownerIdentity":"3a571a9b-03eb-4563-8bc9-cce373be75d5","owner":[],"postedDate":"January 5th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2023-10-16T21:50:45+00:00","versionOfRecord":{"articleIdentity":"rs-2433330","link":"https://doi.org/10.1007/s11118-023-10087-4","journal":{"identity":"potential-analysis","isVorOnly":false,"title":"Potential Analysis"},"publishedOn":"2023-07-22 21:41:11","publishedOnDateReadable":"July 22nd, 2023"},"versionCreatedAt":"2023-01-05 15:23:47","video":"","vorDoi":"10.1007/s11118-023-10087-4","vorDoiUrl":"https://doi.org/10.1007/s11118-023-10087-4","workflowStages":[]},"version":"v1","identity":"rs-2433330","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-2433330","identity":"rs-2433330","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}