{"id":173,"date":"2015-09-16T21:19:20","date_gmt":"2015-09-17T04:19:20","guid":{"rendered":"https:\/\/math.berkeley.edu\/wp\/apde\/?p=173"},"modified":"2015-09-16T21:19:20","modified_gmt":"2015-09-17T04:19:20","slug":"marta-lewicka-university-of-pittsburgh","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/apde\/2015\/09\/16\/marta-lewicka-university-of-pittsburgh\/","title":{"rendered":"Marta Lewicka (University of Pittsburgh)"},"content":{"rendered":"<p>\t\t\t\tSpeaker: Marta Lewicka<\/p>\n<p>Title: &#8220;Convex integration for the Monge-Ampere equation in two dimensions&#8221;.<\/p>\n<p>&nbsp;<\/p>\n<p>Abstract:<\/p>\n<p>We discuss the dichotomy of rigidity vs. flexibility for the $\\mathcal{C}^{1,\\alpha}$ solutions to the Monge-Ampere equation in two dimensions:<\/p>\n<p>\\begin{equation}<\/p>\n<p>{\\mathcal{D}et} \\nabla^2 v := -\\frac 12 \\mbox{curl curl } (\\nabla v \\otimes \\nabla v) = f \\qquad \\mbox{in } \\Omega\\subset\\mathbb{R}^2.<\/p>\n<p>\\end{equation}<\/p>\n<p>Firstly, we show that below the regularity threshold $\\alpha&lt;1\/7$, the very weak $\\mathcal{C}^{1,\\alpha}(\\bar\\Omega)$ solutions to\u00a0 the equation above, (\\ref{MA}), are dense in the set of all continuous functions.<\/p>\n<p>This flexibility statement is a consequence of the convex integration $h$-principle, whereas we directly adapt the iteration method of Nash and Kuiper in order to construct the oscillatory solutions.<\/p>\n<p>Secondly, we prove that the same class of very weak solutions fails the above flexibility in the regularity regime $\\alpha&gt;2\/3$.<\/p>\n<p>Our interest in the regularity of Sobolev solutions to the Monge-Ampere equation is motivated by the variational description of shape formation, which I will also explain in the talk.<\/p>\n<p>&nbsp;\t\t<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Speaker: Marta Lewicka Title: &#8220;Convex integration for the Monge-Ampere equation in two dimensions&#8221;. &nbsp; Abstract: We discuss the dichotomy of rigidity vs. flexibility for the $\\mathcal{C}^{1,\\alpha}$ solutions to the Monge-Ampere equation in two dimensions: \\begin{equation} {\\mathcal{D}et} \\nabla^2 v := -\\frac 12 \\mbox{curl curl } (\\nabla v \\otimes \\nabla v) = f \\qquad \\mbox{in } \\Omega\\subset\\mathbb{R}^2. [&hellip;]<\/p>\n","protected":false},"author":105,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-173","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/posts\/173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/users\/105"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/comments?post=173"}],"version-history":[{"count":0,"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/posts\/173\/revisions"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/media?parent=173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/categories?post=173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/apde\/wp-json\/wp\/v2\/tags?post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}