{"id":612,"date":"2018-05-28T12:48:41","date_gmt":"2018-05-28T12:48:41","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=612"},"modified":"2018-05-28T12:48:41","modified_gmt":"2018-05-28T12:48:41","slug":"hanshudeqiuduichenghuadeyigelizi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=612","title":{"rendered":"\u51fd\u6570\u7684\u7403\u5bf9\u79f0\u5316\u7684\u4e00\u4e2a\u4f8b\u5b50"},"content":{"rendered":"<p>\u5047\u8bbe$u(x)$\u662f$\\mathbb{R}^n$\u4e0a\u4e00\u4e2a(\u5b9e\/\u590d)\u51fd\u6570, \u5b9a\u4e49$u$\u7684\u4e0a\u6c34\u5e73\u96c6\u4e3a<br \/>\n$$<br \/>\n\\mu(t)=|\\{x\\in\\mathbb{R}^m:u(x)>t\\}|.<br \/>\n$$<br \/>\n\u53c8\u5b9a\u4e49$u$\u7684<span class=\"latex_em\">\u7403\u5bf9\u79f0\u91cd\u6392<\/span>(spherically symmetric rearrangement)\u4e3a<br \/>\n$$<br \/>\nu^*(x)=\\sup\\{t:\\mu(t)>|B_{|x|}|\\}<br \/>\n$$<br \/>\n\u5176\u4e2d$B_{|x|}$\u8868\u793a$\\mathbb{R}^m$\u4e2d\u534a\u5f84\u4e3a$|x|$\u7684\u7403\u3002<br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 1<\/span><span class='latex_examp_h'>.<\/span> \u5047\u8bbe$u(x)=4-(x-7)^2$, $x\\in\\mathbb{R}^1$, \u5219$u^*(x)=4-x^2$.<br \/>\n<\/div><br \/>\n<!--more--><\/p>\n<p>\u4e8b\u5b9e\u4e0a, \u76f4\u63a5\u8ba1\u7b97\u5f97\u5230<br \/>\n$$<br \/>\n\\mu(t)=|\\{x\\in\\mathbb{R}:4-(x-7)^2>t\\}|=2\\sqrt{4-t}.<br \/>\n$$<br \/>\n\u800c$|B_{|x|}|=2|x|$. \u56e0\u6b64<br \/>\n$$<br \/>\nu^*(x)=\\sup\\{t:\\mu(t)>|B_{|x|}|\\}=4-|x|^2.<br \/>\n$$<br \/>\n<a href=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2018\/05\/symmetrization.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2018\/05\/symmetrization.png\" alt=\"\" width=\"360\" height=\"223\" class=\"aligncenter size-full wp-image-613\" srcset=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2018\/05\/symmetrization.png 360w, https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2018\/05\/symmetrization-300x186.png 300w\" sizes=\"auto, (max-width: 360px) 100vw, 360px\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5047\u8bbe$u(x)$\u662f$\\mathbb{R}^n$\u4e0a\u4e00\u4e2a(\u5b9e\/\u590d)\u51fd\u6570, \u5b9a\u4e49$u$\u7684\u4e0a\u6c34\u5e73\u96c6\u4e3a $$ \\mu(t&hellip; <a class=\"more-link\" href=\"https:\/\/blog.vanabel.cn\/?p=612\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u51fd\u6570\u7684\u7403\u5bf9\u79f0\u5316\u7684\u4e00\u4e2a\u4f8b\u5b50<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[24,130],"class_list":["post-612","post","type-post","status-publish","format-standard","hentry","category-math","tag-24","tag-qiuduichenghua","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/612","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=612"}],"version-history":[{"count":2,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/612\/revisions"}],"predecessor-version":[{"id":615,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/612\/revisions\/615"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=612"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=612"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=612"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}