{"id":1397,"date":"2025-09-17T20:06:47","date_gmt":"2025-09-18T03:06:47","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/hades\/?p=1397"},"modified":"2025-10-31T22:25:04","modified_gmt":"2025-11-01T05:25:04","slug":"integral-formulas-for-under-overdetermined-differential-operators","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2025\/09\/17\/integral-formulas-for-under-overdetermined-differential-operators\/","title":{"rendered":"Integral formulas for under\/overdetermined differential operators"},"content":{"rendered":"<p>The HADES seminar on Tuesday, <strong>September 23rd<\/strong>, will be at <strong>3<\/strong><strong>:30pm<\/strong>\u00a0in\u00a0<strong>Room 740.<\/strong><\/p>\n<p><strong>Speaker:\u00a0<\/strong>Sung-Jin Oh<\/p>\n<p><strong>Abstract: <\/strong>In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (IH\u00c9S) that introduces a new versatile approach to constructing integral solution operators (i.e., right-inverses up to finite rank operators) for a broad class of\u00a0underdetermined\u00a0operators, including the divergence operator, linearized scalar curvature operator, and the linearized Einstein constraint operator. They are optimally regularizing and, more interestingly, have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). My goal is to (1) describe our approach, (2) demonstrate how it generalizes the well-known construction of Bogovskii, which has proved very useful in fluid dynamics, and (3) explain how it connects underdetermined PDEs with the rich literature on the dual problem on overdetermined differential operators.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Tuesday, September 23rd, will be at 3:30pm\u00a0in\u00a0Room 740. Speaker:\u00a0Sung-Jin Oh Abstract: In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (IH\u00c9S) that introduces a new versatile approach to constructing integral solution operators (i.e., right-inverses up to finite rank operators) for [&hellip;]<\/p>\n","protected":false},"author":95,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[],"class_list":["post-1397","post","type-post","status-publish","format-standard","hentry","category-fall-2025"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1397","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/users\/95"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/comments?post=1397"}],"version-history":[{"count":1,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1397\/revisions"}],"predecessor-version":[{"id":1398,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/1397\/revisions\/1398"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=1397"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=1397"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=1397"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}