{"id":346,"date":"2011-02-15T10:54:31","date_gmt":"2011-02-15T14:54:31","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=346"},"modified":"2011-11-23T08:20:47","modified_gmt":"2011-11-23T13:20:47","slug":"density-11r2","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2011\/02\/15\/density-11r2\/","title":{"rendered":"Density 1\/(1+r^2)"},"content":{"rendered":"<p>Rodrigo Banuelos suggested studying the isoperimetric problem for the radial density 1\/(1+r<sup>2<\/sup>) corresponding to the square root of the Laplacian just as the most important Gaussian density corresponds to the Laplacian itself.<\/p>\n<p><strong>Proposition<\/strong>. Consider <strong>R<\/strong><sup>n<\/sup> with density 1\/(1+<em>r<\/em><sup>2<\/sup>). For <em>n<\/em> &gt; 1 minimizers of perimeter for given volume do not exist: the perimeter can go to zero as the region goes off to infinity. On the line, for more than half the volume the minimizer is a ball about the origin, for less than half, the complement, for exactly half, the ball, its complement, or a half-line. In particular, balls about the origin are minimizing while stable, up to radius 1, with (log density)&#8221; = 2(<em>x<\/em><sup>2<\/sup>-1)\/(<em>x<\/em><sup>2<\/sup>+1)<sup>2<\/sup>.<!--more--><\/p>\n<p><span style=\"font-size: 13.3333px\"><em>Proof.<\/em> For <em>n<\/em>=2, the projected density on the line is \u03c0(1+<em>x<\/em><sup>2<\/sup>)<sup>-1\/2<\/sup>, for which perimeter goes to 0 as an interval of arbitrary fixed volume goes off to infinity. For <em>n<\/em> &gt; 2, balls going off to infinity have perimeter approaching 0. (Consider balls about (0,<em>R<\/em>) of radius <em>R<\/em><sup>\u03b1<\/sup> with \u03b1\u00a0between 2\/n and 2\/(n-1).)<\/span><\/p>\n<p>For <em>n<\/em>=1, since the total volume is finite, minimizers exist and must be half-lines, intervals, or complements [RCBM, Thm. 4.3]. The only stable interval is the ball about the origin of radius at least 1. (There is an unstable equilibrium from -1\/<em>a<\/em> to <em>a<\/em>.) At half the total volume, the half-lines tie the ball; as volume increases, the ball does better for a while because curvature, 2r\/(1+r<sup>2<\/sup>), starts out greater for the ball. That situation eventually reverses, but the half-line never catches the ball, as you can see by considering volume near the maximum (volume \u03c0, perimeter 0), where perimeter of the half-line comes from integrating a larger curvature.<\/p>\n<p>[RCBM] C\u00e9sar Rosales, Antonio Ca\u00f1ete, Vincent Bayle, and Frank Morgan, On the isoperimetric\u00a0problem in Euclidean space with density, Calc. Var. PDE 31 (2008), 27\u201346; <a href=\"http:\/\/arxiv.org\/abs\/math\/0602135\">arXiv.org<\/a> (2006).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Rodrigo Banuelos suggested studying the isoperimetric problem for the radial density 1\/(1+r2) corresponding to the square root of the Laplacian just as the most important Gaussian density corresponds to the Laplacian itself. Proposition. Consider Rn with density 1\/(1+r2). For n &gt; 1 minimizers of perimeter for given volume do not exist: the perimeter can go [&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-346","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/346","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=346"}],"version-history":[{"count":2,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/346\/revisions"}],"predecessor-version":[{"id":828,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/346\/revisions\/828"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=346"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=346"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}