{"id":457,"date":"2013-07-12T16:10:50","date_gmt":"2013-07-12T08:10:50","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=457"},"modified":"2013-07-12T16:10:50","modified_gmt":"2013-07-12T08:10:50","slug":"hopf%e5%bc%95%e7%90%86","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=457","title":{"rendered":"Hopf\u5f15\u7406"},"content":{"rendered":"

\\begin{thm}\u5047\u8bbe$\\Omega\\subset\\R^n$\u662f\u4e00\u4e2a\u6709\u754c\u5f00\u533a\u57df, \u4e14$\\pt\\Omega$\u5149\u6ed1. \u53c8\u8bbe
\n$$
\nL=-a^{ij}(x)D_{ij}+b^i(x)D_i+c(x),
\n$$
\n\u662f$\\Omega$\u4e0a\u7684\u4e00\u81f4\u692d\u5706\u7b97\u5b50($a^{ij}\\xi_i\\xi_j\\geq\\theta|\\xi|^2$), $a^{ij}=a^{ji}$, $b^i$, $c$\u5728$\\bar\\Omega$\u4e0a\u8fde\u7eed. \u5047\u8bbe$u\\in C^2(\\Omega)\\cap C(\\bar\\Omega)$\u4e14$Lu\\geq0$\u5728$\\Omega$\u4e2d\u6210\u7acb. \u82e5\u5b58\u5728\u534a\u5f84\u4e3a$r$\u7684\u5f00\u7403$B\\subset\\Omega$, \u4ee5\u53ca\u4e00\u70b9$x_0\\in\\pt B\\cap\\pt\\Omega$, \u4f7f\u5f97<\/p>\n

    \n
  1. $u$\u5728$x_0$\u5904\u53ef\u5fae,<\/li>\n
  2. $u(x)>u(x_0),\\quad \\forall x\\in \\Omega$,<\/li>\n<\/ol>\n

    \u5219, \u5f53\u4e0b\u5217\u6761\u4ef6\u4e4b\u4e00\u6ee1\u8db3\u65f6,<\/p>\n

      \n
    1. $c\\equiv0$, in $\\Omega$,<\/li>\n
    2. $c\\geq0$, $\\forall x\\in\\Omega$\u4e14$u(x_0)\\leq0$,<\/li>\n
    3. $u(x_0)=0$.<\/li>\n<\/ol>\n

      \u6211\u4eec\u6709, \u5bf9$\\Omega$\u5728$x_0$\u5904\u7684\u5916\u6cd5\u5411\u91cf$\\nu$,<\/p>\n

      $$
      \n\\frac{\\pt u(x_0)}{\\pt\\nu}<0.
      \n$$
      \n\\end{thm}
      \n
      \n\\begin{proof}
      \n\u5206\u6790, \u6211\u4eec\u9700\u8981\u8ba1\u7b97$\\frac{\\pt u}{\\pt\\nu}$\u5728$x_0$\u5904\u7684\u503c, \u4e8e\u662f\u9700\u8981\u8003\u5bdf$x_0$\u7684\u4e00\u4e2a\u90bb\u57df$A\\subset\\Omega$\u65f6$u$\u7684\u503c. \u6ce8\u610f\u5230\u5728$B$\u4e2d\u6709$u(x)>u(x_0)$, \u56e0\u6b64\u6211\u4eec\u7acb\u5373\u53ef\u5f97\u5230(\u7531\u4e8e$\\pt\\Omega$\u5149\u6ed1, \u6545\u6b64\u65f6$B$\u548c$\\pt\\Omega$\u76f8\u5207)
      \n$$
      \n\\frac{\\pt u(x_0)}{\\pt\\nu}\\leq0.
      \n$$
      \n\u4e3a\u4e86\u8bc1\u660e\u4e25\u683c\u4e0d\u7b49\u5f0f\u6210\u7acb, \u6211\u4eec\u9700\u8981\u6784\u9020\u4e00\u4e2a\u51fd\u6570$v$, \u4f7f\u5f97\u5b83\u5728$x_0$\u7684\u4e00\u4e2a\u90bb\u57df$A\\subset\\Omega$\u5185\u6709\u5b9a\u4e49, $u+v$\u5728$x_0$\u5904\u53d6\u5f97$A$\u4e2d\u53d6\u5f97\u6781\u5c0f, \u7279\u522b, \u7531\u4e8e\u6c42\u5bfc\u4e0e\u5e38\u6570\u65e0\u5173, \u7ecf\u8fc7\u9002\u5f53\u5e73\u79fb$v$, \u53ef\u8981\u6c42$(u+v)(x)\\geq(u+v)(x_0)=0$, \u8fd9\u6837\u6211\u4eec\u5c06\u6709
      \n$$
      \n0\\geq\\frac{\\pt(u+v)(x_0)}{\\pt\\nu}=\\frac{\\pt u(x_0)}{\\pt\\nu}+\\frac{\\pt v(x_0)}{\\pt\\nu},
      \n$$
      \n\u56e0\u6b64\u6211\u4eec\u53ea\u9700\u8981\u6c42$$\\frac{\\pt v(x_0)}{\\pt\\nu}>0$$\u5373\u53ef.<\/p>\n

      \u90a3\u4e48\u8fd9\u6837\u7684$v$\u662f\u5426\u5b58\u5728\u5462? \u6ce8\u610f, \u5f31\u6781\u5927\u503c\u539f\u7406\u8868\u660e(\u692d\u5706\u65b9\u7a0b\u7684\u89e3\u7684)\u8fb9\u754c\u503c\u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\u53ef\u63a7\u5236\u5185\u90e8\u7684\u503c, \u56e0\u6b64\u6211\u4eec\u5e0c\u671b$u+v$\u5728$\\pt A$\u7684\u503c$\\geq0$, \u4e14$L(u+v)\\geq0$, $\\forall x\\in A$. \u56de\u5fc6\u5728\u8bc1\u660e\u5f31\u6781\u503c\u539f\u7406<\/a>\u65f6, \u6211\u4eec\u4f5c\u5c0f\u6270\u52a8\u7684\u8f85\u52a9\u51fd\u6570\u662f$-\\eps e^{\\lambda x_1^2}$, \u5f53$\\lambda$\u5145\u5206\u5927\u65f6, \u5b83\u5728$L$\u7684\u4f5c\u7528\u4e0b\u662f\u4e00\u4e2a\u6b63\u9879. \u4e3a\u4e86\u4fbf\u4e8e\u8ba1\u7b97$\\frac{\\pt v}{\\pt\\nu}$\u7684\u503c. \u6211\u4eec\u8fd9\u91cc\u6539\u4e3a\u5c1d\u8bd5\u8003\u5bdf(\u6ce8\u610f\u6211\u4eec\u5047\u8bbe\u4e86$(u+v)(x_0)=0$)<\/p>\n

      $$
      \nv=\\eps e^{\\lambda r^2}-\\eps e^{\\lambda |x-x^0|^2}-u(x_0),
      \n$$
      \n\u5176\u4e2d$x^0$\u662f$B$\u7684\u7403\u5fc3.
      \n\"Hopf
      \n\u6ce8\u610f\u5230
      \n$$
      \n\\frac{\\pt v(x_0)}{\\pt\\nu}=-2\\lambda r\\eps e^{\\lambda|x_0-x^0|^2}<0.
      \n$$
      \n\u8fd9\u6837, \u770b\u8d77\u6765\u6211\u4eec\u5e94\u8be5\u8003\u8651
      \n$$
      \nv=\\eps e^{-\\lambda r^2}-\\eps e^{-\\lambda |x-x^0|^2}-u(x_0),
      \n$$
      \n\u6b64\u65f6\u4fbf\u6709
      \n$$
      \n\\frac{\\pt v(x_0)}{\\pt\\nu}=2\\lambda r\\eps e^{-\\lambda r^2}>0.
      \n$$
      \n\u800c\u8fd9\u4e00\u6539\u53d8\u4e0d\u4f1a\u6539\u53d8$Lv$\u4e2d\u5173\u4e8e$\\lambda$\u7684\u6700\u9ad8\u6b21\u9879\u7684\u7b26\u53f7. \u4e8b\u5b9e\u4e0a, \u82e5\u4ee4$c_0=\\eps e^{-\\lambda r^2}-u(x_0)$, \u5373$v=-\\eps e^{-\\lambda |x-x^0|^2}+c_0$, \u5219:
      \n\\begin{align*}
      \nL(v)&=-\\eps\\left(-a^{ij}D_{ij}+b^iD_i\\right)e^{-\\lambda |x-x^0|^2}+cv\\\\
      \n&=\\eps e^{-\\lambda |x-x^0|^2}\\Bigg\\{
      \n4 \\lambda^2a^{ij}(x_i-x^0_i)(x_j-x_j^0)\\\\
      \n&\\qquad\\qquad\\qquad\\qquad-2\\lambda\\left(\\sum_ia^{ii}-b^i(x_i-x_i^0)\\right)
      \n\\Bigg\\}+cv\\\\
      \n&\\geq\\eps e^{-\\lambda |x-x^0|^2}\\Bigg\\{
      \n4 \\lambda^2\\theta\\|x-x^0\\|^2\\\\
      \n&\\qquad\\qquad\\qquad\\qquad-2\\lambda\\left(\\sum_ia^{ii}-b^i(x_i-x_i^0)\\right)
      \n\\Bigg\\}+cv\\\\
      \n&\\geq\\eps e^{-\\lambda|x-x^0|^2}\\Bigg\\{
      \n4\\theta\\lambda^2\\|x-x^0\\|^2-2\\lambda\\left(\\sum_ia^{ii}-r\\|b\\|\\right)-c
      \n\\Bigg\\}+cc_0\\\\
      \n&\\geq\\eps e^{-\\lambda|x-x^0|^2}\\Bigg\\{
      \n4\\theta\\lambda^2\\|x-x^0\\|^2-2\\lambda\\left(\\sum_ia^{ii}-r\\|b\\|\\right)-c
      \n\\Bigg\\}-cu(x_0)\\\\
      \n\\end{align*}
      \n\u7531\u6b64\u53ef\u89c1, \u5f53$c\\geq0$\u4e14$u(x_0)\\leq0$\u65f6\u6216\u8005$c\\equiv0$\u65f6, \u82e5\u6211\u4eec\u4ee4$A=B\\cap B_{r\/2}(x_0)$\u4e14$\\lambda$\u5145\u5206\u5927\u65f6, \u6709
      \n$$
      \nL(v)\\geq-cu(x_0)\\geq0,\\quad\\forall x\\in A.
      \n$$
      \n\u7531\u6b64, \u5bf9\u4efb\u610f\u7684$\\eps>0$, \u90fd\u5b58\u5728$\\lambda>0$($\\lambda$\u72ec\u7acb\u4e8e$\\eps$), \u4f7f\u5f97
      \n$$
      \nL(u+v)\\geq L(v)\\geq0,\\quad\\forall x\\in A.
      \n$$
      \n\u4e8e\u662f, \u7531\u5f31\u6781\u5927\u503c\u539f\u7406, \u6709
      \n$$
      \n\\min_{\\bar A}(u+v)=\\min_{\\pt A}(-(u+v)^-).
      \n$$
      \n\u53ef\u89c1, \u6211\u4eec\u53ea\u9700\u8bc1\u660e\u5728$\\pt A$\u4e0a\u6709$u+v\\geq0$, \u5219$u+v\\geq0$\u5728$A$\u4e2d\u6210\u7acb.<\/p>\n

      \u6ce8\u610f\u5230$\\pt A=(\\bar A\\cap\\pt B)\\cup(\\pt A\\cap B)$, \u6211\u4eec\u81ea\u7136\u5206\u6210\u4e24\u90e8\u5206\u6765\u8003\u5bdf. \u5728$\\bar A\\cap\\pt B$\u4e0a, \u6211\u4eec\u77e5\u9053$u(x)\\geq u(x_0)$, $v(x)=-u(x_0)$, \u53ef\u89c1\u6b64\u65f6\u6210\u7acb<\/p>\n

      $$
      \nu+v\\geq0.
      \n$$
      \n\u5728$\\pt A\\cap B\\subset\\pt A\\cap \\bar B$\u4e0a, \u6709
      \n$$\\begin{align}
      \nu(x)&>u(x_0),\\\\
      \nv(x)&=\\eps e^{\\lambda r^2}-\\eps e^{\\lambda |x-x^0|^2}-u(x_0)\\notag\\\\
      \n&\\geq \\eps\\left(e^{-\\lambda r^2}-e^{-\\lambda r^2\/4}\\right)-u(x_0),
      \n\\end{align}$$
      \n\u53ef\u89c1\u6b64\u65f6\u6211\u4eec\u53ef\u53d6$\\eps$\u5145\u5206\u5c0f, \u4f7f\u5f97,
      \n$$
      \nu+v>0.
      \n$$
      \n\u8fd9\u6837, \u6211\u4eec\u5c31\u5b8c\u6210\u4e86\u5b9a\u7406\u5728$c\\geq0$\u4e14$u(x_0)\\leq0$\u60c5\u5f62\u7684\u8bc1\u660e.<\/p>\n

      \u81f3\u4e8e$u(x_0)=0$. \u8003\u5bdf$\\tilde L=L+c^-$, \u6b64\u65f6\u6709<\/p>\n

      $$
      \n\\tilde a^{ij}=a^{ij},\\quad \\tilde b_i=b_i,\\quad \\tilde c=c+c^-=c^+\\geq0,
      \n$$
      \n\u4e14
      \n$$
      \n\\tilde L u=Lu+c^-u\\geq Lu\\geq0,
      \n$$
      \n\u8fd9\u662f\u56e0\u4e3a, \u7531\u5047\u8bbe$u(x)> u(x_0)=0$\u5728$\\Omega$\u4e2d\u6210\u7acb. \u4e8e\u662f, \u6211\u4eec\u53ea\u9700\u5bf9$\\tilde L$\u5e94\u7528\u524d\u9762\u7684\u7ed3\u679c\u5373\u53ef.
      \n\\end{proof}<\/p>\n

      \u6700\u540e, \u6211\u4eec\u5c06\u4ee5\u4e00\u4e2a\u6ce8\u8bb0\u6765\u7ed3\u675fHopf\u5f15\u7406\u7684\u7b14\u8bb0.<\/p>\n

      \\begin{rem}
      \n\\ref{thm:1}\u7684\u6761\u4ef6:$u(x)>u(x_0)$\u5728$\\Omega$\u4e2d\u6210\u7acb, \u53ef\u4ee5\u7a0d\u5fae\u51cf\u5f31\u70b9(\u6b63\u5982\u6709\u7684\u4e66\u4e0a\u6240\u8bf4), \u6211\u4eec\u53ef\u4ee5\u5047\u8bbe$u(x)>u(x_0)$\u5728$B$\u4e2d\u6210\u7acb, \u6b64\u65f6\u8bc1\u660e\u57fa\u672c\u4e0d\u53d8, \u552f\u4e00\u8981\u6ce8\u610f\u7684\u662f\u5728$\\pt A\\cap B\\subset\\pt A\\cap \\bar B$\u4e0a, \u6211\u4eec\u53ea\u80fd\u5f97\u5230$u(x)\\geq u(x_0)$(\u4e0d\u518d\u662f\u4e25\u683c\u5927\u4e8e), \u8fd9\u5c06\u5bfc\u81f4$\\eps$\u7684\u5b58\u5728\u6027\u6709\u95ee\u9898. \u4f46\u662f\u6211\u4eec\u53ea\u9700\u5c06\u8003\u5bdf\u533a\u57df\u6362\u6210\u73af\u72b6\u533a\u57df$A=B_{r}(x^0)\\setminus \\bar B_{r\/2}(x^0)$\u5373\u53ef.
      \n\\end{rem}<\/p>\n","protected":false},"excerpt":{"rendered":"

      \\begin{thm}\u5047\u8bbe$\\Omega\\subset\\R^n$\u662f\u4e00\u4e2a\u6709\u754c\u5f00\u533a\u57df …<\/p>\n

      Hopf\u5f15\u7406<\/span> \u9605\u8bfb\u66f4\u591a »<\/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":[508],"class_list":["post-457","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-hop"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/457","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=457"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/457\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=457"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=457"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=457"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}