{"id":1116,"date":"2011-12-08T21:41:49","date_gmt":"2011-12-08T13:41:49","guid":{"rendered":"http:\/\/vanabel.sinaapp.com\/?p=1116"},"modified":"2011-12-08T21:41:49","modified_gmt":"2011-12-08T13:41:49","slug":"an-min-max-problem-in-riemannian-geometry","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=1116","title":{"rendered":"An Min-Max Problem in Riemannian Geometry"},"content":{"rendered":"<p>\\begin{prob}<br \/>\nLet $\\Omega$ be a connected open subset of $\\R^n$ and $u:\\Omega\\to\\R$ be a $C^\\infty$ function. Call $\\Delta$ the <strong>Laplace operator<\/strong><br \/>\n\\[<br \/>\n\\Delta u=\\sum_{i=1}^n\\frac{\\partial^2 u}{\\partial (x^i)^2}.<br \/>\n\\]<br \/>\nIf $u$ satisfies $\\Delta u\\geq0$, prove that $u$ cannot achieve a strict local maximum at any interior point of $\\Omega$.<br \/>\n\\end{prob}<br \/>\n<!--more--><br \/>\n\\begin{answer}<br \/>\nHere is the first answer&#8230;<br \/>\n\\end{answer}<br \/>\n\\begin{answer}<br \/>\nHere is another answer&#8230;<br \/>\n\\end{answer}<br \/>\n\\begin{prob}<br \/>\nLet $(M,g)$ be a complete Riemannian manifold without boundary (possibly noncompact) and $u\\mathpunct{:}M\\to\\R$ be a $C^\\infty$ function. Call $\\Delta$ the <strong>Laplace operator<\/strong> of $g$<br \/>\n\\[<br \/>\n\\Delta u=g^{ij}\\frac{\\partial^2 u}{\\partial x^i\\partial x^j}-g^{ij}\\Gamma_{ij}^k\\frac{\\partial u}{\\partial x^k}.<br \/>\n\\]<br \/>\nwhere $\\Gamma_{ij}^k$ are the <strong>Christoffel symbols<\/strong> of the Levi-Civita connection of $g$. If $u$ satisfies $\\Delta u\\geq 0$, prove that $u$ cannot achieve a strict local maximum at any point of $M$.<br \/>\n\\end{prob}<br \/>\n\\begin{answer}<br \/>\nThe first answer for problem 2<br \/>\n\\end{answer}<br \/>\n\\begin{answer}<br \/>\nThe second answer for problem 2<br \/>\n\\end{answer}<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\\begin{prob} Let $\\Omega$ be a connected &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/lttt.vanabel.cn\/?p=1116\"> <span class=\"screen-reader-text\">An Min-Max Problem in Riemannian Geometry<\/span> \u9605\u8bfb\u66f4\u591a &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[557,608],"class_list":["post-1116","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-min-max-problem","tag-riemannian-geometry"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1116"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1116\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1116"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1116"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}