{"id":405,"date":"2016-06-17T07:07:34","date_gmt":"2016-06-17T07:07:34","guid":{"rendered":"http:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/?p=405"},"modified":"2016-06-17T07:07:34","modified_gmt":"2016-06-17T07:07:34","slug":"holder-continuity-for-solutions-of-eikonal-equations","status":"publish","type":"post","link":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/2016\/06\/17\/holder-continuity-for-solutions-of-eikonal-equations\/","title":{"rendered":"H\u00f6lder continuity for solutions of eikonal equations"},"content":{"rendered":"

Ermal\u00a0Feleqi (Cardiff, UK)<\/em><\/p>\n

I will talk about results\u00a0 on H\u00f6lder continuity of viscosity solutions of eikonal PDEs<\/p>\n

|\u2207X u| = f\u00a0\u00a0\u00a0 in \u03a9<\/p>\n

u = 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 on \u2202\u03a9<\/p>\n

structured on possibly\u00a0 degenerate and\u00a0 nonsmooth systems of vector\u00a0 fields X = (X1 , . . . , Xp ) .<\/p>\n

The\u00a0 typical result\u00a0 goes as follows:\u00a0 if the\u00a0 given vector\u00a0 fields satisfy\u00a0 H\u00f6r- mander\u2019s\u00a0 condition of at most\u00a0 step\u00a0 k at each\u00a0 interior point of \u03a9 and\u00a0 if at each\u00a0 boundary point of \u03a9 a Lie bracket of degree\u00a0 at most\u00a0 k can by found not\u00a0 tangential (i.e.,\u00a0 transversal) to \u2202\u03a9, then\u00a0 the\u00a0 solution\u00a0 of (P)\u00a0 is (1\/k)- H\u00f6lder continuous. The proof relies on representing u as the value function of an optimal control\u00a0 problem:\u00a0 actually, when f \u2261 1, as the minimum time to\u00a0 reach\u00a0 the\u00a0 exterior of \u03a9 by\u00a0 X -trajectories, that is, concatenations of a finite number of integral curves\u00a0 of the\u00a0 vector\u00a0 fields \u00b1Xi <\/sub>, i = 1, . . . , p.<\/p>\n

The\u00a0 smoothness assumptions on vector\u00a0 fields and\u00a0 \u2202\u03a9 are\u00a0 reduced\u00a0 to\u00a0 a \u201cbare\u00a0 minimum\u201d.\u00a0\u00a0\u00a0 For\u00a0 vector\u00a0 fields\u00a0 this\u00a0 is made\u00a0 possible\u00a0 by\u00a0 introducing a\u00a0 set-valued notion\u00a0\u00a0 of iterated Lie\u00a0 bracket which\u00a0 makes\u00a0 sense\u00a0 for\u00a0 quite nonsmooth vector\u00a0 fields.\u00a0\u00a0 Concerning the\u00a0 regularity of \u03a9, we require\u00a0 for it to satisfy\u00a0 an exterior cone condition. Then\u00a0 the\u00a0 transversality condition is expressed\u00a0 by requiring that all the\u00a0 vectors\u00a0 of a set-valued bracket point toward an exterior cone.\u00a0 When\u00a0 \u03a9 is of class C 1\u00a0 or possesses Bony normals the\u00a0 transversality condition can\u00a0 be phrased in more\u00a0 natural terms\u00a0 (every vector\u00a0 of the\u00a0 bracket should\u00a0 not\u00a0 be orthogonal to the\u00a0 normal). We\u00a0 can\u00a0 cover\u00a0 also\u00a0 boundaries with\u00a0 isolated points.\u00a0 At those\u00a0 isolated points\u00a0 the\u00a0 trensversality condition is expressed\u00a0 by requiring that H\u00f6rman- der\u2019s condition be satisfied\u00a0 therein.<\/p>\n

Joint work with Franco Rampazzo, Martino Bardi\u00a0 and Pierpaolo Soravia<\/p>\n

Back to Programme<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Ermal\u00a0Feleqi (Cardiff, UK) I will talk about results\u00a0 on H\u00f6lder continuity of viscosity solutions of eikonal PDEs |\u2207X u| = f\u00a0\u00a0\u00a0 in \u03a9 u = 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 on \u2202\u03a9 structured on possibly\u00a0 degenerate and\u00a0 nonsmooth systems of vector\u00a0 fields X = (X1 , . . . , Xp ) . The\u00a0 typical result\u00a0 goes as follows:\u00a0 … Continue reading H\u00f6lder continuity for solutions of eikonal equations<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-405","post","type-post","status-publish","format-standard","hentry","category-abstracts"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/posts\/405","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/comments?post=405"}],"version-history":[{"count":0,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/posts\/405\/revisions"}],"wp:attachment":[{"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/media?parent=405"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/categories?post=405"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/ntnlpde16\/wp-json\/wp\/v2\/tags?post=405"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}