{"id":303,"date":"2016-12-13T12:27:48","date_gmt":"2016-12-13T12:27:48","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=303"},"modified":"2016-12-21T05:29:44","modified_gmt":"2016-12-21T05:29:44","slug":"ruodiaoheyingzhaodeoula-lagelangri","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=303","title":{"rendered":"\u5f31\u8c03\u548c\u6620\u7167\u7684\u6b27\u62c9—\u62c9\u683c\u6717\u65e5"},"content":{"rendered":"

1. \u5f31\u8c03\u548c\u6620\u7167<\/a><\/span>\n\n\u5047\u8bbe$(M,g)$, $(N,h)$\u662f\u4e24\u4e2a\u9ece\u66fc\u6d41\u5f62, \u4e14$N \\hookrightarrow \\mathbb{R}^K $. \u5b9a\u4e49
\n\\[
\n H^1(M,N):=\\left\\{ u\\in L_{\\mathrm{loc}}^1(M,\\mathbb{R}^{K+1}):\\int_{M}|\\nabla u|^2<+\\infty \\& u(x)\\in N \\text{a.e.} \\right\\}.\n\\]\n\u5bf9$u\\in H^1(M,N)$, \u5b9a\u4e49\u80fd\u91cf\u6cdb\u51fd\n\\[\n E(u)=\\int_{M}L(x,u,\\nabla u)=\\int_M\\frac{1}{2}g^{\\alpha\\beta}\\left\\langle\\frac{\\partial u}{\\partial x^\\alpha},\\frac{\\partial u}{\\partial x^\\beta}\\right\\rangle.\n\\]\n

\u5b9a\u4e49 1<\/span>.<\/span> \u6211\u4eec\u79f0$u\\in H^1(M,N)$\u662f\u82e5\u8c03\u548c\u6620\u7167<\/span>, \u5982\u679c$u$\u662f$E(u)$\u5728\u5982\u4e0b\u53d8\u5206\u7684\u4e34\u754c\u70b9:\n \\[\n \\lim_{t\\to0}\\frac{E(P(u+tv))-E(u)}{t}=0,\\quad\\forall v\\in H_0^1(M,\\mathbb{R}^{K})\\bigcap L^\\infty(M,\\mathbb{R}^{K}),\n \\]\n \u5176\u4e2d$H_0^1$\u662f$C_0^\\infty$\u5728$H^1$\u4e2d\u7684\u5b8c\u5907\u5316, \u800c$P$\u662f$N$\u7684\u6700\u8fd1\u70b9\u6295\u5c04.\n<\/div><\/p>\n

\u4e0b\u9762, \u6211\u4eec\u6765\u8ba1\u7b97\u5176\u6b27\u62c9—\u62c9\u683c\u6717\u65e5\u65b9\u7a0b.
\n

\u5f15\u7406 2<\/span>.<\/span> $u\\in H^1(M,N)$\u662f\u82e5\u8c03\u548c\u6620\u7167\u5f53\u4e14\u4ec5\u5f53\u5982\u4e0b\u65b9\u7a0b\u5728\u5f31\u610f\u4e49\u4e0b\u6210\u7acb
\n \\[
\n \\Delta_g u+g^{\\alpha\\beta}A(u)\\left( \\frac{\\partial u}{\\partial x^\\alpha},\\frac{\\partial u}{\\partial x^\\beta} \\right)=0,
\n \\]
\n \u5373\u5bf9\u4efb\u4f55$v\\in H_0^1(M,\\mathbb{R}^{K})\\bigcap L^\\infty(M,\\mathbb{R}^K)$, \u6709
\n \\[
\n \\int_{M}-g^{\\alpha\\beta}(x) \\left\\langle \\frac{\\partial u}{\\partial x^\\alpha},\\frac{\\partial v}{\\partial x^\\beta} \\right\\rangle +g^{\\alpha\\beta}(x)\\left\\langle A(u)\\left( \\frac{\\partial u}{\\partial x^\\alpha},\\frac{\\partial u}{\\partial x^\\beta} \\right),v \\right\\rangle=0,
\n \\]
\n \u5176\u4e2d$A$\u662f$N \\hookrightarrow \\mathbb{R}^K$\u7684\u7b2c\u4e8c\u57fa\u672c\u578b.
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u6211\u4eec\u76f4\u63a5\u8ba1\u7b97\u5f97\u5230
\n \\begin{align*}
\n \\left.\\frac{d}{dt}\\right\\rvert_{t=0}\\left( \\frac{1}{2}|\\nabla \\Pi(u+tv)|^2 \\right)
\n &=\\left\\langle\\nabla u,\\left.\\frac{d}{dt}\\right\\rvert_{t=0}(\\nabla\\Pi(u+tv))\\right\\rangle\\\\
\n &=\\left\\langle \\nabla u,\\nabla [D\\Pi(u)](v) \\right\\rangle\\\\
\n &=\\left\\langle \\nabla u,\\nabla v^\\top \\right\\rangle,
\n \\end{align*}
\n \u56e0\u6b64\u7531$v\\in H_0^1(M, \\mathbb{R}^K)$\u5e76\u5229\u7528\u5206\u90e8\u79ef\u5206\u77e5
\n \\[
\n \\left.\\frac{d}{dt}\\right\\rvert_{t=0}\\int_{M}\\frac{1}{2}|\\nabla \\Pi(u+tv)|^2
\n =\\int_M\\left\\langle \\nabla u,\\nabla v^\\top \\right\\rangle
\n =-\\int_{M}\\left\\langle \\Delta u,v^\\top \\right\\rangle
\n =-\\int_{M}\\left\\langle [\\Delta u]^\\top,v \\right\\rangle.
\n \\]
\n \u73b0\u5728\u5047\u8bbe$\\set{\\nu_i(u)}$\u662f$u$\u5904$T_uN^\\perp$\u7684\u5e7a\u6b63\u57fa\u5e95, \u5219\u6ce8\u610f\u5230$\\left\\langle \\nabla u,\\nu_i(u) \\right\\rangle=0$, \u6211\u4eec\u77e5\u9053
\n \\begin{align*}
\n \\left\\langle \\Delta u,\\nu_i(u) \\right\\rangle &= \\mathrm{div}\\left( \\left\\langle \\nabla u,\\nu_i(u) \\right\\rangle \\right)-\\left\\langle \\nabla u,D(\\nu_i(u)) \\right\\rangle
\n =-\\left\\langle \\nabla u, [D\\nu_i]( \\nabla u) \\right\\rangle\\\\
\n &=-\\left\\langle \\nabla u, D_{\\nabla u}\\nu_i\\right\\rangle
\n =-A^i(u)(\\nabla u,\\nabla u).
\n \\end{align*}
\n \u6700\u540e\u4e00\u4e2a\u7b49\u53f7\u4e2d, \u6211\u4eec\u7528\u4e86\u7b2c\u4e8c\u57fa\u672c\u578b$A$\u7684\u5b9a\u4e49. \u8fd9\u6837, \u6211\u4eec\u5f97\u5230
\n \\[
\n \\Delta u-[\\Delta u]^\\top=[\\Delta u]^\\perp=\\left\\langle \\Delta u,\\nu_i \\right\\rangle\\nu_i=-A^i(u)(\\nabla u,\\nabla u)\\nu_i=-A(u)(\\nabla u,\\nabla u).
\n \\]
\n \u8fd9\u8868\u660e
\n \\[
\n \\left.\\frac{d}{dt}\\right\\rvert_{t=0}\\int_{M}\\frac{1}{2}|\\nabla \\Pi(u+tv)|^2=-\\int_{M}\\left\\langle [\\Delta u]^\\top,v \\right\\rangle
\n =-\\int_M \\left\\langle \\Delta u+A(u)(\\nabla u,\\nabla u), v \\right\\rangle.
\n \\]
\n \u7531\u6b64\u5373\u5f97\u5176\u6b27\u62c9—\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\u4e3a
\n \\[
\n \\Delta u+A(u)(\\nabla u,\\nabla u)=0.
\n \\]
\n \u6700\u540e\u6ce8\u610f\u5230$A$\u7684\u53cc\u7ebf\u6027\u6027, \u4ee5\u53ca$\\nabla u=g^{\\alpha\\delta}\\frac{\\partial u}{\\partial x^\\alpha}\\frac{\\partial }{\\partial x^\\delta}$
\n \\[
\n A(u)(\\nabla u,\\nabla u)=g^{\\alpha\\beta}A(u)\\left( \\frac{\\partial u}{\\partial x^\\alpha},\\frac{\\partial u}{\\partial x^\\beta} \\right).
\n \\]
\n \u4e8b\u5b9e\u4e0a, \u4e00\u4e2a\u4ed4\u7ec6\u7684\u8ba1\u7b97\u8868\u660e
\n \\begin{align*}
\n \\left\\langle \\Delta u, \\nu_i(u) \\right\\rangle_N \\nu_i(u)
\n &=\\left\\langle \\mathrm{div}_g(\\nabla_g u),\\nu_i(u) \\right\\rangle_N \\nu_i(u)\\\\
\n &=\\mathrm{div}_g\\left\\langle \\nabla_gu,\\nu_i(u) \\right\\rangle_N\\nu_i(u)
\n -g^{\\alpha\\beta}\\left\\langle \\nabla_{\\frac{\\partial}{\\partial x^\\alpha}}u,D_{\\frac{\\partial}{\\partial x^\\beta}}\\nu_i(u) \\right\\rangle_{ N}\\nu_i(u)\\\\
\n &=-g^{\\alpha\\beta}\\left\\langle \\frac{\\partial u}{\\partial x^\\alpha},(D\\nu_i)(\\nabla_{\\frac{\\partial}{\\partial x^\\beta}}u) \\right\\rangle_{N}\\nu_i(u)\\\\
\n &=-g^{\\alpha\\beta}A(u)\\left( \\frac{\\partial u}{\\partial x^\\alpha},\\frac{\\partial u}{\\partial x^\\beta} \\right).
\n \\end{align*}
\n<\/div>
\n\u7279\u522b\u5730, \u5f53$N=S^n$\u65f6, \u6211\u4eec\u6709
\n
\u63a8\u8bba 3<\/span>.<\/span> \u5047\u8bbe$u\\in H^1(M,S^n)$\u662f\u82e5\u8c03\u548c\u6620\u7167, \u5219\u5176\u6b27\u62c9—\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\u662f
\n \\[
\n \\Delta_g u+|\\nabla_gu|^2u=0.
\n \\]
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u82e5$N=S^n \\hookrightarrow \\mathbb{R}^{n+1}$, \u5219\u6211\u4eec\u53ef\u4ee5\u5c06$S^n$\u7684\u5916\u6cd5\u5411$\\nu=u$\u5f84\u5411\u5ef6\u62d3\u4e3a$\\mathbb{R}^{n+1}$\u4e2d\u7684\u5355\u4f4d\u5411\u91cf\u573a$\\tilde u(y)=u(y\/|y|)$. \u8fd9\u6837, \u8ba1\u7b97\u5176\u7b2c\u4e8c\u57fa\u672c\u578b\u6709, \u5bf9$X,Y\\in T_uS^n$,
\n \\[
\n A(u)(X,Y)= \\left\\langle D_X\\nu,Y \\right\\rangle \\nu = \\left\\langle \\bar D_X \\tilde u,Y \\right\\rangle \\nu
\n = \\left\\langle (d\\tilde u)(X),Y \\right\\rangle u
\n =\\left\\langle X,Y \\right\\rangle u.
\n \\]
\n \u5176\u4e2d, \u6211\u4eec\u9996\u5148\u5229\u7528\u4e86\u5d4c\u5165\u5b50\u6d41\u5f62\u8bf1\u5bfc\u8054\u7edc\u7684\u57fa\u672c\u5173\u7cfb$D_X\\nu=[\\bar D_X\\nu]^\\top$, \u8fd9\u91cc$\\bar D$\u662f\u6b27\u6c0f\u7a7a\u95f4$\\mathbb{R}^{n+1}$\u4e2d\u7684Levi-Civita\u8054\u7edc, \u5176\u5b9e\u5b83\u5c31\u662f\u65b9\u5411\u5bfc\u6570(\u53ef\u4ee5\u9a8c\u8bc1\u5176\u5ea6\u91cf\u76f8\u5bb9\u6027\u4e0e\u65e0\u6320\u6027). \u5373$D_X \\tilde u=(d\\tilde u)(X)$. \u53c8\u6ce8\u610f\u5230, $\\tilde u(y)=y$\u5176\u5b9e\u5c31\u662f\u8be5\u7535\u5904\u7684\u4f4d\u7f6e\u5411\u91cf. \u800c\u5173\u4e8e\u4f4d\u7f6e\u5411\u91cf, \u6211\u4eec\u5bb9\u6613\u5f97\u5230$d(\\tilde u)(X)=X$.<\/p>\n

\u7531\u6b64, \u6211\u4eec\u77e5\u9053
\n \\[
\n g^{\\alpha\\beta}A(u)\\left( \\frac{\\partial u}{\\partial x^\\alpha},\\frac{\\partial u}{\\partial x^\\beta} \\right)=g^{\\alpha\\beta}\\left\\langle \\frac{\\partial u}{\\partial x^\\alpha},\\frac{\\partial u}{\\partial x^\\beta} \\right\\rangle u
\n =|\\nabla_g u|^2u.
\n \\]
\n<\/div><\/p>\n","protected":false},"excerpt":{"rendered":"

1. \u5f31\u8c03\u548c\u6620\u7167 \u5047\u8bbe$(M,g)$, $(N,h)$\u662f\u4e24\u4e2a\u9ece\u66fc\u6d41\u5f62, \u4e14$N \\hookrig… \u7ee7\u7eed\u9605\u8bfb\u5f31\u8c03\u548c\u6620\u7167\u7684\u6b27\u62c9—\u62c9\u683c\u6717\u65e5<\/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":[57,58],"class_list":["post-303","post","type-post","status-publish","format-standard","hentry","category-math","tag-ruodiaoheyingzhao","tag-oula-lagelangri","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/303","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=303"}],"version-history":[{"count":6,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/303\/revisions"}],"predecessor-version":[{"id":336,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/303\/revisions\/336"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=303"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=303"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=303"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}