{"id":103,"date":"2020-10-05T14:29:57","date_gmt":"2020-10-05T12:29:57","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/mprats\/?p=103"},"modified":"2025-07-09T15:24:09","modified_gmt":"2025-07-09T13:24:09","slug":"marti-prats-and-xavier-tolsa-the-two-phase-problem-for-harmonic-measure-in-vmo","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/mprats\/2020\/10\/05\/marti-prats-and-xavier-tolsa-the-two-phase-problem-for-harmonic-measure-in-vmo\/","title":{"rendered":"Mart\u00ed Prats and Xavier Tolsa: The two-phase problem for harmonic measure in VMO"},"content":{"rendered":"\n<div class=\"wp-block-group is-nowrap is-layout-flex wp-container-core-group-is-layout-8f761849 wp-block-group-is-layout-flex\">\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link wp-element-button\" href=\"https:\/\/arxiv.org\/abs\/1904.00751\" target=\"_blank\" rel=\"noreferrer noopener\">arXiv<\/a><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link wp-element-button\" href=\"https:\/\/doi.org\/10.1007\/s00526-020-01760-2\" target=\"_blank\" rel=\"noreferrer noopener\">Calc. Var. PDE<\/a><\/div>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let \\(\\Omega^+\\subset\\mathbb R^{n+1}\\) be an NTA domain and let \\(\\Omega^-= \\mathbb R^{n+1}\\setminus \\overline{\\Omega^+}\\) be an NTA domain as well. Denote by \\(\\omega^+\\) and \\(\\omega^-\\) their respective harmonic measures. Assume that \\(\\Omega^+\\) is a \\(\\delta\\)-Reifenberg flat domain, for some \\(\\delta&gt;0\\) small enough. In this paper we show that \\(\\log\\frac{d\\omega^-}{d\\omega^+}\\in VMO(\\omega^+)\\) if and only if \\(\\Omega^+\\) is vanishing Reifenberg flat, \\(\\omega^+\\) has big pieces of uniformly rectifiable measures, and the inner unit normal of \\(\\Omega^+\\) has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let \\(\\Omega^+\\subset\\mathbb R^{n+1}\\) be an NTA domain and let \\(\\Omega^-= \\mathbb R^{n+1}\\setminus \\overline{\\Omega^+}\\) be an NTA domain as well. Denote by \\(\\omega^+\\) and \\(\\omega^-\\) their respective harmonic measures. Assume that \\(\\Omega^+\\) is a \\(\\delta\\)-Reifenberg flat domain, for some \\(\\delta&gt;0\\) small enough. In this paper we show that \\(\\log\\frac{d\\omega^-}{d\\omega^+}\\in VMO(\\omega^+)\\) if and only if \\(\\Omega^+\\) is &hellip; <a href=\"https:\/\/mat.uab.cat\/web\/mprats\/2020\/10\/05\/marti-prats-and-xavier-tolsa-the-two-phase-problem-for-harmonic-measure-in-vmo\/\" class=\"more-link\">Continua la lectura de <span class=\"screen-reader-text\">Mart\u00ed Prats and Xavier Tolsa: The two-phase problem for harmonic measure in VMO<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[95],"tags":[39],"class_list":["post-103","post","type-post","status-publish","format-standard","hentry","category-papers-en","tag-papers-en"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/103","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/users\/53"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/comments?post=103"}],"version-history":[{"count":6,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/103\/revisions"}],"predecessor-version":[{"id":451,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/103\/revisions\/451"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/media?parent=103"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/categories?post=103"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/tags?post=103"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}