{"id":367,"date":"2011-02-21T16:19:46","date_gmt":"2011-02-21T20:19:46","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=367"},"modified":"2021-10-20T06:16:33","modified_gmt":"2021-10-20T11:16:33","slug":"density-expra","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2011\/02\/21\/density-expra\/","title":{"rendered":"Density exp(\u00b1r^a)"},"content":{"rendered":"<p>An interesting class of densities for the isoperimetric problem on <strong>R<\/strong><sup>n<\/sup> is exp(\u00b1<em>r<sup>a<\/sup><\/em>). The <a href=\"http:\/\/blogs.williams.edu\/Morgan\/2010\/04\/03\/the-log-convex-density-conjecture\/\">Log-convex Density Conjecture<\/a> says that spheres about the origin are isoperimetric if and only if 0\u2264<em>a<\/em>\u22641 with the minus sign or\u00a0<em>a<\/em> \u2265 1 with the plus sign, known for <em>a<\/em>=2 [Borell; see\u00a0<a href=\"http:\/\/arxiv.org\/abs\/math\/0602135\">Rosales\u00a0et al.<\/a><a href=\"http:\/\/arxiv.org\/abs\/math\/0602135\">]<\/a> and in <strong>R<\/strong><sup>2<\/sup> for <em>a<\/em>\u22652\u00a0<a href=\"http:\/\/www.springerlink.com\/content\/e6w88754478rp15h\/\">[Maurmann-Morgan]<\/a>. Henceforth we&#8217;ll focus on <strong>R<\/strong><sup>2<\/sup>.<!--more--><\/p>\n<p><strong>Density exp(<\/strong><strong><em>r<\/em><sup><em>a<\/em><\/sup>) on R<sup>2<\/sup>.<\/strong> Circles about the origin are stable unless 0&lt;<em>a<\/em>&lt;1. They are conjectured isoperimetric for \u00a0<em>a<\/em>\u22651, proven for <em>a<\/em>\u22652 <a href=\"http:\/\/www.springerlink.com\/content\/e6w88754478rp15h\/\">[Maurmann-Morgan]<\/a> and for a&gt;1 for large balls <a href=\"http:\/\/arxiv.org\/abs\/1002.1829\">(Kolesnikov-Zhdanov<\/a>,\u00a0Rmk. 6.9, generalized by Howe [H]), and supported by some numerical evidence for <em>a<\/em>=1 [L, Prop. 6.8].\u00a0For <em>a<\/em>&lt;0, the density is decreasing to 1 at infinity with severe nonconvex blow-up at the origin, and minimizers probably disappear at infinity; yet a circle <em>S<\/em> about the origin is probably perimeter minimizing in competition with any other curve <em>C<\/em> such that <em>C<\/em>\u2013<em>S<\/em> bounds net area 0, as follows for small circles by Howe [H].<\/p>\n<p>For 0&lt;<em>a<\/em>&lt;1 circles about the origin are unstable, but minimizers exist <a href=\"http:\/\/arxiv.org\/abs\/math\/0602135\">[Rosales\u00a0et al.<\/a>,\u00a0Thm. 3.5].\u00a0They are probably transcendental ovals containing the origin.<\/p>\n<p><strong>Density exp(\u2013r<sup><em>a<\/em><\/sup>) on R<sup>2<\/sup>.<\/strong> When 0&lt;<em>a<\/em>&lt;1, the density has severe nonconvexity at the origin but decreases rapidly to 0 at infinity, yielding finite total area and hence existence of isoperimetric curves; a circle about the origin is probably isoperimetric.\u00a0In the famous case of Gaussian density (<em>a<\/em>=2), straight lines are uniquely isoperimetric. As <em>a<\/em> decreases from 2 to 1, numerical studies\u00a0<a href=\"http:\/\/arxiv.org\/abs\/1002.1829\">[Kolesnikov-Zhdanov<\/a>,\u00a0Sect. 5]\u00a0show a nice transition from a straight line to a curved line to a large closed curve to a circle; as <em>a<\/em> increases from 2, the line curves in the other direction.<\/p>\n<p>In the interesting case of <em>a<\/em> &lt; 0, the density increases to 1 at infinity. Isoperimetric curves exist [MP]. They probably pass through the origin, where the density vanishes.<\/p>\n<p>[H] \u00a0Sean Howe, The Log-Convex Density Conjecture and vertical surface area in warped products,<a href=\"http:\/\/arxiv.org\/abs\/1107.4402\">arXiv.org<\/a>\u00a0(2011).<\/p>\n<p>[L] \u00a0Yifei Li, Michael Mara, Isamar Rosa Plata, and Elena Wikner, Tiling with penalties and isoperimetry with density,\u00a0<a href=\"http:\/\/www.rose-hulman.edu\/mathjournal\/v13n1.php\">Rose-Hulman Und. Math. J. 13 (1) (2012)<\/a>.<br \/>\n[MP] \u00a0Frank Morgan and Aldo Pratelli, Existence of isoperimetric regions in <strong>R<\/strong><sup>n<\/sup> with density, <a href=\"http:\/\/arxiv.org\/abs\/1111.5160\">arXiv.org<\/a> (2011).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>An interesting class of densities for the isoperimetric problem on Rn is exp(\u00b1ra). The Log-convex Density Conjecture says that spheres about the origin are isoperimetric if and only if 0\u2264a\u22641 with the minus sign or\u00a0a \u2265 1 with the plus sign, known for a=2 [Borell; see\u00a0Rosales\u00a0et al.] and in R2 for a\u22652\u00a0[Maurmann-Morgan]. Henceforth we&#8217;ll focus [&hellip;]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-367","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/367","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=367"}],"version-history":[{"count":7,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/367\/revisions"}],"predecessor-version":[{"id":3183,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/367\/revisions\/3183"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=367"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}