The fractional obstacle problem in $\\mathbb R^n$ with obstacle $\\varphi\\in C^\\infty(\\mathbb{R}^n)$ can be written as<\/p>\n\n\n\n
\\[\\min\\{(-\\Delta)^s u , u-\\varphi\\} = 0,\\quad\\textrm{in }\\quad\\mathbb{R}^n. \\] The set $\\{u = \\varphi\\} \\subset \\mathbb{R}^n$ is called the contact set, and its boundary is the free boundary, an unknown of the problem.<\/p>\n\n\n\n
The free boundary for the fractional obstacle problem can be divided between two subsets: regular points (around which the free boundary is smooth, and is $n-1$ dimensional) and degenerate points. The set of degenerate points, even for smooth obstacles, can be very large (for example, with infinite $\\mathcal{H}^{n-1}$ measure). In joint work with X. Ros-Oton, we show, however, that generically solutions to the fractional obstacle problem have a lower-dimensional degenerate set. That is, for almost every solution (in an appropriate sense), the set of degenerate points is lower-dimensional.<\/p>","protected":false},"excerpt":{"rendered":"
The fractional obstacle problem in $\\mathbb R^n$ with obstacle $\\varphi\\in C^\\infty(\\mathbb{R}^n)$ can be written as \\[\\min\\{(-\\Delta)^s u , u-\\varphi\\} = 0,\\quad\\textrm{in }\\quad\\mathbb{R}^n. \\] The set $\\{u = \\varphi\\} \\subset \\mathbb{R}^n$ is called the contact set, and its boundary is the free boundary, an unknown of the problem. The free boundary for the fractional obstacle problem can be divided between two subsets: regular points<\/p>","protected":false},"author":13,"featured_media":6035,"parent":0,"template":"","meta":{"advgb_blocks_editor_width":"","advgb_blocks_columns_visual_guide":""},"tags":[],"event_types":[26],"event_times":[],"acf":[],"featured_img":"https:\/\/aims-cameroon.org\/wp-content\/uploads\/sites\/6\/2021\/10\/Xavier-AIMS-Virtual_Colloquium_page-0001.jpg","coauthors":[],"author_meta":{"author_link":"https:\/\/aims-cameroon.org\/fr\/author\/george\/","display_name":"george"},"relative_dates":{"created":"Publi\u00e9 5 ans il y a","modified":"Mis \u00e0 jour 5 ans il y a"},"absolute_dates":{"created":"Publi\u00e9 le octobre 28, 2021","modified":"Mise \u00e0 jour le octobre 28, 2021"},"absolute_dates_time":{"created":"Publi\u00e9 le octobre 28, 2021 3:18 ","modified":"Mise \u00e0 jour le octobre 28, 2021 3:18 "},"featured_img_caption":"","tax_additional":{"post_tag":{"linked":[],"unlinked":[],"slug":"post_tag","name":"\u00c9tiquettes"},"event_types":{"linked":["Workshop<\/a>"],"unlinked":["Workshop<\/span>"],"slug":"event_types","name":"Event Type"},"event_times":{"linked":[],"unlinked":[],"slug":"event_times","name":"Event Time"}},"series_order":"","_links":{"self":[{"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/event\/6034"}],"collection":[{"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/event"}],"about":[{"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/types\/event"}],"author":[{"embeddable":true,"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":2,"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/event\/6034\/revisions"}],"predecessor-version":[{"id":6037,"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/event\/6034\/revisions\/6037"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/media\/6035"}],"wp:attachment":[{"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/media?parent=6034"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/tags?post=6034"},{"taxonomy":"event_types","embeddable":true,"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/event_types?post=6034"},{"taxonomy":"event_times","embeddable":true,"href":"https:\/\/aims-cameroon.org\/fr\/wp-json\/wp\/v2\/event_times?post=6034"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}