.<\/span> \u5047\u8bbe$\\xi$\u662f\u652f\u5728$|x|<2$\u7684\u622a\u65ad\u51fd\u6570, \u4e14$\\xi\\equiv1$, $\\forall |x|<1$. \u5728\u65b9\u7a0b\u4e24\u8fb9\u540c\u4e58$\\phi\\in C_0^\\infty$, \u5206\u90e8\u79ef\u5206\u6709
\n$$
\n0=\\int e^{-\\frac{|x|^2}{4}}\\nabla u\\cdot\\nabla\\phi\\rd x,
\n$$
\n\u7279\u522b, \u4ee4$\\phi=\\xi u$,
\n$$\\begin{align*}
\n0&=\\int e^{-\\frac{|x|^2}{4}}(|\\nabla u|^2\\xi+u\\nabla u\\cdot\\nabla \\xi)\\rd x\\\\
\n &=\\int e^{-\\frac{|x|^2}{4}}\\left(|\\nabla u|^2\\xi+|\\nabla u|\\xi^{\\frac{1}{2}}\\cdot \\frac{u \\nabla \\xi}{\\xi^{\\frac{1}{2}}}\\right)\\\\
\n &\\geq \\int e^{-\\frac{|x|^2}{4}}\\left(
\n |\\nabla u|^2\\xi-\\frac{1}{2}|\\nabla u|^2\\xi-\\frac{1}{2}u^2\\frac{|\\nabla \\xi|^2}{\\xi}
\n \\right)\\\\
\n &= \\int e^{-\\frac{|x|^2}{4}}\\left(
\n \\frac{1}{2}|\\nabla u|^2\\xi-\\frac{1}{2}u^2\\frac{|\\nabla\\xi|^2}{\\xi}
\n \\right),
\n\\end{align*}$$
\n\u73b0\u5728, \u4ee4$\\eta=\\xi^2$, \u5229\u7528\u4e0a\u5f0f, \u6709
\n$$\\begin{align*}
\n\\int_{|x| < r} e^{-\\frac{|x|^2}{4}}|\\nabla u|^2&\\leq\\int e^{-\\frac{|x|^2}{4}}\\eta|\\nabla u|^2\\rd x\\leq\\int e^{-\\frac{|x|^2}{4}}u^2\\frac{|\\nabla\\eta|^2}{\\eta}\\\\
\n&=4\\int e^{-\\frac{|x|^2}{4}}u^2|\\nabla\\xi|^2\\\\
\n&\\leq\\frac{C}{r^2}\\int_{r < |x| < 2r} e^{-\\frac{|x|^2}{4}}u^2\\rd x.
\n\\end{align*}$$
\n\u6700\u540e\u4e00\u6b65, \u6211\u4eec\u5229\u7528\u4e86\u622a\u65ad\u51fd\u6570\u7684\u6027\u8d28.
\n<\/div>
\n\u975e\u5e38\u611f\u8c22\u8001\u5e08\u63d0\u4f9b\u7684\u8bc1\u660e.<\/p>\n","protected":false},"excerpt":{"rendered":"\u547d\u9898 1. \u5047\u8bbe$u$\u662f\u65b9\u7a0b $$ 0=\\Delta u-\\frac{1}{2}x\\cdot \\nabla u… \u7ee7\u7eed\u9605\u8bfb\u6807\u51c6\u7684\u692d\u5706\u7406\u8bba:\u4e00\u4e2a\u80fd\u91cf\u4e0d\u7b49\u5f0f<\/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":[133,112,134],"class_list":["post-628","post","type-post","status-publish","format-standard","hentry","category-math","tag-jichu","tag-tuoyuanfangcheng","tag-nengliangbudengshi","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/628","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=628"}],"version-history":[{"count":1,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/628\/revisions"}],"predecessor-version":[{"id":629,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/628\/revisions\/629"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=628"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=628"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=628"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}