{"id":398,"date":"2013-07-05T22:07:07","date_gmt":"2013-07-05T14:07:07","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=398"},"modified":"2013-07-05T22:07:07","modified_gmt":"2013-07-05T14:07:07","slug":"%e5%bc%b1%e6%9e%81%e5%80%bc%e5%8e%9f%e7%90%86","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=398","title":{"rendered":"\u5f31\u6781\u503c\u539f\u7406"},"content":{"rendered":"

\u4e0d\u77e5\u4e3a\u5565, \u770b\u4e66\u4e0a\u8001\u89c9\u5f97\u6ca1\u5199\u6e05\u695a. \u6211\u8bd5\u4e3e\u51e0\u4e2a\u5730\u65b9:<\/p>\n

    \n
  1. \u5173\u4e8e\u53d6\u6781\u9650, \u5728\u5047\u8bbe\u692d\u5706\u7b97\u5b50\u534a\u6b63\u65f6\u8981\u7528\u6270\u52a8<\/strong>\u6765\u8f6c\u5316\u4e3a\u4e25\u683c\u6b63\u7684\u60c5\u5f62, \u4f46\u662f\u6700\u540e\u53d6\u6781\u9650\u4e3a\u4ec0\u4e48\u80fd\u591f\u5f97\u5230\u7ed3\u8bba?<\/li>\n
  2. \u5173\u4e8e$c\\geq0$\u7684\u60c5\u5f62, \u5f88\u591a\u4e66\u4e0a\u662f\u8003\u8651$\\Omega^+=\\set{x\\in\\Omega|u(x)<0}$\u8fd9\u6837\u7684\u533a\u57df, \u5728$\\Omega^+$\u4e0a\u7528$c\\equiv0$\u7684\u60c5\u51b5, \u4f46\u662f\u6700\u540e\u4e3a\u4ec0\u4e48\u4f1a\u4ece$\\Omega^+$\u4ee5\u53ca$\\partial\\Omega^+$\u8f6c\u5230$\\Omega$, $\\partial\\Omega$\u4e0a\u7ed3\u8bba\u6210\u7acb?<\/li>\n<\/ol>\n

    <\/p>\n

    \n

    \u6211\u4eec\u5148\u505a\u4e00\u4e9b\u57fa\u672c\u5047\u8bbe:<\/p>\n

      \n
    1. $\\Omega\\subset\\R^n$\u662f\u4e00\u4e2a\u6709\u754c\u5f00\u533a\u57df(\u4ece\u800c\u8fde\u901a);<\/li>\n
    2. $u\\in C^2(\\Omega)\\cap C(\\bar\\Omega)$;<\/li>\n
    3. $Lu:=-a^{ij}(x)u_{ij}+b^i(x)u_i+c(x)u$\u79f0\u4e3a\u662f\u692d\u5706\u7b97\u5b50, \u5982\u679c\u5bf9\u4efb\u610f\u7684$x\\in\\Omega$, \u90fd\u6709$a^{ij}\\xi_i\\xi_j\\geq\\theta\\|\\xi\\|^2$, \u8fd9\u91cc$\\theta>0$, \u800c$\\xi=(\\xi_1,\\xi_2,\\ldots,\\xi_n)\\in\\R^n$.<\/li>\n
    4. $a^{ij}=a^{ji}$, $b^i$, $c$\u90fd\u662f$\\Omega$\u4e0a\u7684\u8fde\u7eed\u51fd\u6570.<\/li>\n<\/ol>\n

      \\begin{lem}
      \n\u5047\u8bbe\u5728$\\Omega$\u4e2d$Lu>0$, \u4e14$c\\equiv0$, \u5219$u$\u5728$\\Omega$\u4e2d\u4e0d\u80fd\u53d6\u5f97\u6781\u5c0f\u503c.
      \n\\end{lem}
      \n\\begin{proof}\u4e8b\u5b9e\u4e0a, \u53cd\u8bbe$x_0\\in\\Omega$\u5904, $u$\u53d6\u5f97\u6781\u5c0f\u503c. \u5219\u7531\u4e8e$u\\in C^2(\\Omega)$, \u6211\u4eec\u901a\u8fc7\u6cf0\u52d2\u5c55\u5f00\u77e5\u9053
      \n$$
      \n\\nabla u(x_0)=0,\\quad D^2u(x_0)\\geq0.
      \n$$
      \n\u73b0\u5728, \u6211\u4eec\u5c06\u7528\u7ebf\u6027\u4ee3\u6570\u91cc\u9762\u4e00\u4e2a\u57fa\u672c\u4e8b\u5b9e\u6765\u8bf4\u660e\u77db\u76fe: \u5047\u8bbe$A=(-a^{ij})\\leq0$, $B=D^2u(x_0)\\geq0$, \u5219\u6709$\\tr(AB)\\leq0$, \u8fd9\u6837\u8bb0\u5f97\u5230$Lu(x_0)\\leq0$, \u4e0e\u9898\u8bbe\u77db\u76fe.<\/p>\n

      \u5173\u4e8e\u7ebf\u6027\u4ee3\u6570\u7684\u8fd9\u4e2a\u4e8b\u5b9e, \u53ea\u9700\u6ce8\u610f\u5230$B$\u662f\u5b9e\u5bf9\u79f0\u7684, \u4ece\u800c\u5b58\u5728\u6b63\u4ea4\u9635$O$, \u4f7f\u5f97
      \n$$
      \nO^{-1}BO
      \n=\\pmatrix{\\lambda_1&0&\\cdots&0\\\\
      \n0&\\lambda_2&\\cdots&0\\\\
      \n\\vdots&\\vdots&&\\vdots\\\\
      \n0&0&\\cdots&\\lambda_n}:=\\Lambda
      \n$$
      \n\u56e0\u6b64 $B=O\\Lambda O^{-1}$,
      \n$$
      \n\\tr(AB)=\\tr(AO\\Lambda O^{-1})=\\tr(O^{-1}AO\\Lambda)=\\tr(O^TAO\\Lambda).
      \n$$
      \n\u7531\u4e8e$A\\leq0$, \u4ece\u800c\u6309\u5b9a\u4e49\u5373\u77e5$O^TAO\\leq0$, \u7279\u522b\u5176\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u90fd\u5c0f\u4e8e\u7b49\u4e8e$0$, \u7531\u6b64\u6613\u89c1$\\tr(AB)\\leq0$(\u56e0\u4e3a$B\\geq0\\Rightarrow\\lambda_i\\geq0$).
      \n\\end{proof}
      \n\u5982\u679c\u6211\u4eec\u628a$c\\equiv0$\u7684\u6761\u4ef6\u51cf\u5f31\u4e3a$c\\geq0$, \u5219\u7ed3\u8bba\u5e94\u53d8\u6210:
      \n\\begin{lem}
      \n\u5047\u8bbe\u5728$\\Omega$\u4e2d$Lu>0$, \u4e14$c\\geq0$, \u5219$u$\u5728$\\Omega$\u4e2d\u4e0d\u80fd\u53d6\u5f97\u975e\u6b63\u7684\u6781\u5c0f\u503c, \u5373, \u5176\u975e\u6b63\u6781\u5c0f\u503c\u53ea<\/strong>\u5728\u8fb9\u754c\u53d6\u5f97; \u4e5f\u5373\u8981\u4e48\u5176\u6781\u5c0f\u503c\u53ea<\/strong>\u5728\u8fb9\u754c\u53d6\u5f97, \u8981\u4e48(\u5728\u5185\u90e8\u53d6\u5f97\u65f6)\u5176\u6781\u5c0f\u503c\u5927\u4e8e\u96f6.
      \n\\end{lem}
      \n\\begin{proof}
      \n\u4e8b\u5b9e\u4e0a, \u5047\u8bbe$L_0u=Lu-c(x)u>-c(x)u$, \u4ece\u800c\u53cd\u8bbe\u80fd\u5728$\\Omega$\u4e2d$x_0$\u5904\u53d6\u5f97\u975e\u6b63\u6781\u5c0f\u503c, \u5219\u5728$x_0$\u5904\u6211\u4eec\u4ecd\u6709$L_0u(x_0)>0$, \u8fd9\u6837\u7531\u524d\u9762\u7684\u8bc1\u660e\u77e5\u9053$x_0$\u4e0d\u80fd\u662f$u$\u7684\u6781\u5c0f\u503c\u70b9, \u77db\u76fe.
      \n\\end{proof}
      \n\u4e0b\u9762, \u6211\u4eec\u901a\u8fc7\u6270\u52a8\u5c06\u7ed3\u679c\u6539\u8fdb\u5230$Lu\\geq0$\u7684\u60c5\u5f62, \u53ef\u4ee5\u770b\u5230\u7ed3\u8bba\u4e8b\u5b9e\u4e0a\u8981\u76f8\u5e94\u7684\u51cf\u5f31<\/strong>.
      \n\\begin{thm}
      \n\u5047\u8bbe\u5728$\\Omega$\u4e2d$Lu\\geq0$, \u4e14$c\\equiv0$, \u5219
      \n$$
      \n\\min_{\\bar\\Omega}u=\\min_{\\pt\\Omega}u.
      \n$$
      \n\\end{thm}
      \n\\ref{thm:1}\u76f8\u5bf9\u4e8e\\ref{lem:1}\u6761\u4ef6\u51cf\u5f31\u4e86, \u6211\u4eec\u5e94\u8be5\u6ce8\u610f\u5230\u5176\u5b9e, \u5b83\u7684\u7ed3\u8bba\u4e5f\u51cf\u5f31\u4e86.
      \n\\begin{rem}
      \n\u5f53$Lu>0$\u65f6, \u7531\\ref{lem:1}\u6211\u4eec\u77e5\u9053, \u6b64\u65f6\u5b58\u5728$x_0\\in\\pt\\Omega$, \u4f7f\u5f97$u(x_0)=\\min_{\\bar\\Omega}u$, \u56e0\u6b64\u81ea\u7136\u6709$\\min_{\\bar\\Omega}u=\\min_{\\pt\\Omega}u$. \u4f46\u53cd\u8fc7\u6765, $\\min_{\\bar\\Omega}u=\\min_{\\pt\\Omega}u$, \u5e76\u4e0d\u80fd\u8bf4\u660e$u$\u5728$\\Omega$\u5185\u90e8\u4e0d\u80fd\u53d6\u5f97\u5176\u5728$\\bar\\Omega$\u4e0a\u7684\u6781\u5c0f\u503c. \u8fd9\u5c31\u662f\u8bf4\u5f53\u6761\u4ef6\u51cf\u5f31\u65f6, \u7ed3\u8bba\u5176\u5b9e\u4e5f\u76f8\u5e94\u7684\u51cf\u5f31\u4e86.
      \n\\end{rem}
      \n\\begin{proof}
      \n\u5b9a\u7406\u7684\u8bc1\u660e\u6709\u4e2a\u6280\u5de7, \u5c31\u662f\u8003\u5bdf\u8f85\u52a9\u51fd\u6570$u_\\eps=u-\\eps e^{\\lambda x_1}$, \u8fd9\u91cc$\\lambda>0$\u5f85\u5b9a. \u76f4\u63a5\u8ba1\u7b97\u6709,
      \n$$\\begin{align*}
      \nLu_\\eps&=Lu-\\eps L(e^{\\lambda x_1})
      \n\\geq-\\eps e^{\\lambda x_1}\\left(-\\lambda^2a^{11}+\\lambda b^1 \\right)\\\\
      \n&=\\eps e^{\\lambda x_1}\\left(\\lambda^2a^{11}-\\lambda b^1 \\right)\\\\
      \n&\\geq\\eps e^{\\lambda x_1}\\left(\\lambda^2\\theta-\\lambda \\|b\\|_{L^\\infty} \\right),
      \n\\end{align*}$$
      \n\u53ef\u89c1, \u6211\u4eec\u53ef\u4ee5\u9009\u53d6$\\lambda$\u5145\u5206\u5927(\u6ce8\u610f, $\\lambda$\u72ec\u7acb\u4e8e$\\eps$), \u4f7f\u5f97$Lu_\\eps>0$\u5728$\\Omega$\u4e0a\u6210\u7acb. \u8fd9\u6837\u7531\\ref{lem:1}\u77e5,
      \n$$
      \n\\min_{\\bar\\Omega}u\\geq \\min_{\\bar\\Omega}u_\\eps=\\min_{\\pt\\Omega}u_\\eps
      \n\\geq \\min_{\\pt\\Omega}u+\\min_{\\pt\\Omega}\\left(-\\eps e^{\\lambda x_1}\\right)
      \n= \\min_{\\pt\\Omega}u-\\eps\\max_{\\pt\\Omega}e^{\\lambda x_1}.
      \n$$
      \n\u8fd9\u6837, \u6211\u4eec\u4ee4$\\eps\\to0$\u4fbf\u5f97\u5230
      \n$$
      \n\\min_{\\bar\\Omega}u=\\min_{\\pt\\Omega}u.
      \n$$
      \n\\end{proof}
      \n\u5b8c\u5168\u7c7b\u4f3c\u5730, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230$c\\geq0$\u7684\u60c5\u5f62.
      \n\\begin{thm}
      \n\u5047\u8bbe\u5728$\\Omega$\u4e2d$Lu\\geq0$, \u4e14$c\\geq0$, \u5219
      \n$$
      \n\\min_{\\bar\\Omega}u\\geq\\min_{\\pt\\Omega}(-u^-).
      \n$$
      \n\\end{thm}
      \n\u8fd9\u91cc, \u6211\u4eec\u4ecd\u7136\u6709\u6761\u4ef6\u51cf\u5f31, \u7ed3\u8bba\u5e94\u8be5\u76f8\u5e94\u51cf\u5f31\u7684\u89c2\u5bdf.
      \n\\begin{rem}\u5f53$Lu>0$\u65f6, \u7531\\ref{lem:2}\u81ea\u7136\u8574\u542b\u7740$\\min_{\\bar\\Omega}u\\geq\\min_{\\pt\\Omega}(-u^-)$. \u4e8b\u5b9e\u4e0a, \u53cd\u8bbe$\\min_{\\bar\\Omega}u<\\min_{\\pt\\Omega}(-u^-)\\leq0$, \u5219\u5b58\u5728$x_0\\in\\bar\\Omega$, \u4f7f\u5f97$u(x_0)=\\min_{\\bar\\Omega}u<0$. \u82e5$x_0\\in\\Omega$, \u5219\u8bf4\u660e$u$\u5728$\\Omega$\u5185\u90e8\u53d6\u5f97\u6781\u5c0f\u4e14\u662f\u975e\u6b63\u7684, \u77db\u76fe. \u56e0\u800c$x_0\\in\\pt\\Omega$, \u4f46\u7531$u(x_0)<0$\u77e5$u(x_0)=-u^-(x_0)\\geq \\min_{\\pt\\Omega}(-u^-)$, \u8fd9\u4e5f\u662f\u77db\u76fe\u7684.<\/p>\n

      \u4f46\u662f, \u53cd\u8fc7\u6765, $\\min_{\\bar\\Omega}u\\geq\\min_{\\pt\\Omega}(-u^-)$\u5e76\u4e0d\u80fd\u63a8\u51fa$u$\u5728$\\Omega$\u5185\u4e0d\u80fd\u53d6\u5f97\u975e\u6b63\u6781\u5c0f. \u4f8b\u5982$u\\equiv0$\u5c31\u662f\u4e00\u4e2a\u53cd\u4f8b.
      \n\\end{rem}
      \n$c\\geq0$\u60c5\u5f62\u4e0b\u5b9a\u7406\u7684\u8bc1\u660e\u5e76\u65e0\u5b9e\u8d28\u533a\u522b. \u56e0\u4e3a\u8fd9\u65f6\u6211\u4eec\u6709
      \n$$
      \nLu_\\eps\\geq\\eps e^{\\lambda x_1}\\left(\\lambda^2\\theta-\\lambda \\|b\\|_{L^\\infty} -c\\right)
      \n\\geq\\eps e^{\\lambda x_1}\\left(\\lambda^2\\theta-\\lambda \\|b\\|_{L^\\infty} -\\|c\\|_{L^\\infty}\\right),
      \n$$
      \n\u6211\u4eec\u4ecd\u53ef\u53d6$\\lambda$\u5145\u5206\u5927(\u72ec\u7acb\u4e8e$\\eps$)\u4f7f\u5f97$Lu_\\eps>0$\u5728$\\Omega$\u4e0a\u6210\u7acb. \u63a5\u4e0b\u6765\u7684\u6b65\u9aa4\u540c\u524d.<\/p>\n

      \n

      \u6700\u540e, \u6211\u4eec\u4f5c\u70b9\u601d\u8003:<\/p>\n

        \n
      1. \u5bf9$Lu\\leq0$\u7684\u60c5\u5f62, \u6211\u4eec\u53ef\u901a\u8fc7\u8003\u5bdf$L(-u)\\geq0$\u6765\u5f97\u5230\u76f8\u5e94\u7684\u7ed3\u8bba, \u5373\u628a\u524d\u9762\u7ed3\u8bba\u4e2d\u7684$u$\u6362\u6210$-u$\u5373\u53ef.<\/li>\n
      2. \u6211\u4eec\u80fd\u5426\u786e\u5b9e\u6784\u9020\u4e00\u4e2a\u4f8b\u5b50\u8bf4\u660e$Lu\\geq0$\u4e14$c\\geq0$\u65f6, $u$\u80fd\u5728$\\Omega$\u5185\u53d6\u5f97\u6b63\u7684\u6781\u5c0f\u503c?<\/li>\n<\/ol>\n

        Written with StackEdit<\/a>.<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

        \u4e0d\u77e5\u4e3a\u5565, \u770b\u4e66\u4e0a\u8001\u89c9\u5f97\u6ca1\u5199\u6e05\u695a. \u6211\u8bd5\u4e3e\u51e0\u4e2a\u5730\u65b9: \u5173\u4e8e\u53d6\u6781\u9650, \u5728\u5047\u8bbe\u692d\u5706\u7b97 …<\/p>\n

        \u5f31\u6781\u503c\u539f\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":[169],"class_list":["post-398","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-169"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/398","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=398"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/398\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=398"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=398"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=398"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}