{"id":4290,"date":"2014-01-13T11:53:56","date_gmt":"2014-01-13T03:53:56","guid":{"rendered":"https:\/\/lttt.blog.ustc.edu.cn\/?p=4290"},"modified":"2014-01-13T11:53:56","modified_gmt":"2014-01-13T03:53:56","slug":"%e5%87%a0%e4%bd%95%e5%88%86%e6%9e%90%e4%b8%ad%e7%9a%84%e5%8f%98%e5%88%86%e9%97%ae%e9%a2%98%e4%b8%8e%e6%96%b9%e6%b3%95","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=4290","title":{"rendered":"\u51e0\u4f55\u5206\u6790\u4e2d\u7684\u53d8\u5206\u95ee\u9898\u4e0e\u65b9\u6cd5"},"content":{"rendered":"

\u8fd9\u662f\u4e01\u4f1f\u5cb3\u9662\u58eb\u7684\u4e00\u4e2aTalk, \u539f\u6587\u94fe\u63a5\u5728\u5176\u4e3b\u9875\u4e0a\u6709: \u51e0\u4f55\u5206\u6790\u4e2d\u7684\u53d8\u5206\u95ee\u9898\u4e0e\u65b9\u6cd5<\/a>.<\/p>\n

\\section{\u5386\u53f2\u7684\u56de\u987e:1960\u5e74\u4ee5\u524d}
\n\u53d8\u5206\u6cd5\u6709\u5f88\u957f\u7684\u5386\u53f2\uff0c\u5982\u679c\u4ece\u6b27\u62c9\u548c\u62c9\u683c\u6717\u65e5\u63d0\u51fa\u4ee5\u4ed6\u4eec\u7684\u540d\u5b57\u547d\u540d\u7684\u53d8\u5206\u65b9\u7a0b\u7b97\u8d77\uff0c\u81f3\u4eca\u5df1\u6709250\u5e74\u7684\u5386\u53f2\u3002\u5728\u5f00\u59cb\u7684\u65f6\u5019\uff0c\u53d8\u5206\u6cd5\u7684\u521b\u7acb\u548c\u5e94\u7528\u4e3b\u8981\u662f\u56f4\u7ed5\u7269\u7406\u5b66(\u529b\u5b66\uff0c\u5149\u5b66\uff0c\u5929\u6587\u5b66\u7b49\u7b49 )\u4e2d\u7684\u5404\u79cd\u53d8\u5206\u95ee\u9898\u3002\u6bd4\u5982\uff0c\u4e0e\u62c9\u666e\u62c9\u65af\u65b9\u7a0b\u76f8\u8054\u7cfb \u7684Dirichlet\u539f\u7406\u5c31\u662f\u5728\u7814\u7a76\u5f15\u529b\u6216\u7535\u573a\u7684\u4f4d\u52bf\u65f6\u63d0\u51fa\u7684\u3002
\n\u53d8\u5206\u6cd5\u5bf9\u4e8e\u51e0\u4f55\u7684\u5e94\u7528\u5728\u65e9\u671f\u4e3b\u8981\u662f\u5bf9\u66f2\u9762\u4e0a\u7684\u6d4b\u5730\u7ebf\u548c\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u7ed9\u5b9a\u8fb9\u754c\u7684\u6781\u5c0f\u66f2\u9762(Plateau\u95ee\u9898 )\u7684\u7814\u7a76\u3002\u4f46\u5728\u5f88\u957f\u65f6\u671f\u5185\u4ec5\u9650\u4e8e\u4e00\u4e9b\u7279\u6b8a\u60c5\u5f62\uff0c\u6ca1\u6709\u91cd\u8981\u8fdb\u5c55\u3002
\n\u76f4\u5230\u4e0a\u4e16\u7eaa\u65e9\u671f\uff0c\u4e3a\u4e86\u7814\u7a76\u66f2\u9762\u4e0a\u7684\u6d4b\u5730\u7ebf\u7684\u4e2a \u6570\uff0cMorse(20-30\u5e74\u4ee3)\u548c\u4fc4\u56fd\u6570\u5b66\u5bb6(40\u5e74\u4ee3)\u5206\u522b\u5efa\u7acb\u4e86Morse\u548c Ljusternik-Schnirelman\u7406\u8bba\u3002\u5176\u4e2d\uff0cMorse\u7406\u8bba\u4e0d\u4ec5\u5bf9\u53d8\u5206\u95ee\u9898\u7684\u89e3\u7684\u4e2a\u6570\u4f30\u8ba1\u6709\u8bb8\u591a\u5e94\u7528\u800c\u4e14\u5728\u6d41\u5f62\u7684\u62d3\u6251\u95ee\u9898\u6709\u91cd\u8981\u5e94\u7528\u3002
\n1930\u5e74\u4ee3\uff0cDouglas\u548cRado\u5206\u522b\u7ed9\u51fa\u4e86Plateau\u95ee\u9898(\u5706\u76d8\u60c5\u5f62)\u7684\u89e3\u7b54\u3002Douglas\u56e0\u6b64\u83b7\u5f97\u4e861936\u5e74\u7684\u7b2c\u4e00\u5c4aFields\u5956\u3002\u8fd9\u4e2a\u672c\u6765\u662f\u9762\u79ef\u6cdb\u51fd\u6781\u5c0f\u5316\u7684\u95ee\u9898\uff0c\u88ab\u5316\u4e3aDirichlet\u79ef\u5206\u6781\u5c0f\u5316\u95ee\u9898;\u7136\u540e\u5229\u7528Dirichlet\u79ef\u5206\u7684\u5171\u5f62\u4e0d\u53d8\u6027<\/em>\uff0c\u5706\u76d8\u81ea\u8eab\u7684\u5171\u5f62\u53d8\u6362<\/em>\uff0c\u4ee5\u53ca\u5206\u6790\u7684\u6280\u5de7\uff0c\u6765\u6784\u9020\u5206\u6790\u4e0a\u53ef\u63a7\u7684\u6781\u5c0f\u5316\u5e8f\u5217\u3002\u8fd9\u53ef\u4ee5\u8bf4\u662f\u5728\u53d8\u5206\u95ee\u9898\u4e2d\u5e94\u7528\u51e0\u4f55\u65b9\u6cd5(\u5171\u5f62\u51e0\u4f55)\u7684\u7b2c\u4e00\u4e2a\u4f8b\u5b50\u3002\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u5178\u578b\u7684\u51e0\u4f55\u5206\u6790\u7684\u65b9\u6cd5\uff0c\u5373\u628a\u51e0\u4f55\u4e0e\u5206\u6790\u81ea\u7136\u5730\u7ed3\u5408\u8d77\u6765\u7684\u65b9\u6cd5\u3002<\/p>\n

$$\\begin{cases}
\nu:D^2\\to\\R^n, u(\\pt D)=\\Gamma\\subset\\R^n\\\\
\n\\Delta u=0\\\\
\n|u_x|^2-|u_y|^2=u_x\\cdot u_y=0\\quad (\\text{Conformality})
\n\\end{cases}$$
\n\\section{\u5386\u53f2\u7684\u56de\u987e:\u4e0a\u4e16\u7eaa60\u5e74\u4ee3}
\n60\u5e74\u4ee3\u4ece\u8bb8\u591a\u65b9\u9762\u6765\u770b\u662f\u73b0\u4ee3\u6570\u5b66\u53d1\u5c55\u7684\u4e00\u4e2a\u91cd\u8981\u65f6\u671f\u3002\u4f46\u6211\u4eec\u53ea\u80fd\u6d89\u53ca\u51e0\u4e2a\u6709\u5173\u7684\u91cd\u8981\u5de5\u4f5c\u3002<\/p>\n

    \n
  1. Palais\u628aMorse\u7406\u8bba\u548cLjusternik-Schnirelman\u7406\u8bba\u63a8\u5e7f\u5230\u65e0\u7a77\u7ef4\u7684Banach\u6d41\u5f62\uff0c\u5e76\u540cSmale\u4e00\u8d77\u63d0\u51fa\u4e86Palais-Smale<\/strong>\u7d27\u6027\u6761\u4ef6\u3002(1963-66)<\/li>\n
  2. Eells\u4e0eSampson\u53d1\u8868\u4e86\u5f00\u521b\u6027\u7684\u8bba\u6587\u201dHarmonic mappings of Riemannian manifolds\u201d(1964)\u3002\u4ece\u6b64\uff0c\u8c03\u548c\u6620\u5c04<\/strong>\u4f5c\u4e3a\u4e00\u4e2a\u51e0\u4f55\u53d8\u5206\u95ee\u9898\u5f00\u59cb\u5f15\u8d77\u4eba\u4eec\u5173\u6ce8.
    \n

    Let $(M; g)$ be a Riemannian manifold, and $N\\subset \\R^N$ be a<\/p>\n

    submanifold of $\\R^N$. A mapping $u_0 : M\\to N$ is called harmonic<\/p>\n

    map i\ufb00 it is a critical point of the following energy functional<\/p>\n

    $$
    \nE(u)=\\int_M|\\nabla u|^2_g\\rd V_g
    \n$$<\/p>\n

    where in local coordinates<\/p>\n

    $$\\begin{gather*}
    \n|\\nabla u|^2_g=g^{ij}\\inner{\\frac{\\pt u}{\\pt x^i},\\frac{\\pt u}{\\pt x^j}}\\\\
    \n\\rd V_g=\\sqrt{\\det(g_{ij})}\\rd x
    \n\\end{gather*}$$<\/p>\n

    The Euler-Lagrange operator of $E(u)$ is called \u2018tension \ufb01eld\u2019,<\/p>\n

    denoted $\\tau(u)$, which is elliptic, and if $N = S^{N-1}$ is the round<\/p>\n

    sphere,<\/p>\n

    $$
    \n\\tau (u)=\\Delta_g u+|\\nabla u|_g^2 u.
    \n$$<\/p><\/blockquote>\n<\/li>\n<\/ol>\n

    \u4e3a\u4e86\u5f97\u5230\u8c03\u548c\u6620\u5c04\u7684\u5b58\u5728\u6027\uff0cEells-Sampson\u5f15\u5165\u4e86\u201d\u8c03\u548c\u6620\u5c04\u7684\u70ed\u65b9\u7a0b<\/strong>\u201c:<\/p>\n

    $$
    \nu_t = \u03c4(u)
    \n$$<\/p>\n

    \u4f5c\u4e3a$E(u)$\u7684\u4e00\u4e2a\u201d\u8d1f\u68af\u5ea6\u6d41\u201d\u3002
    \n\u4ed6\u4eec\u8bc1\u660e \u5982\u679c$N$\u7684\u201d\u622a\u9762\u66f2\u7387$\u22640$\u201d,\u5219\u4efb\u7ed9$u_0 \\in C^1(M,N)$, \u70ed\u65b9\u7a0b\u4ee5$u_0$\u4e3a\u521d\u503c\u7684\u89e3\u957f\u65f6\u95f4\u5b58\u5728\uff0c\u5e76\u4e14\u5f53$t \u2192 0$\u65f6\u6536\u655b\u5230\u4e0e$u_0$\u540c\u4f26\u7684\u8c03\u548c\u6620\u5c04$u_\\infty$\u3002
    \n\u73b0\u5728\uff0c\u70ed\u6d41\u65b9\u6cd5<\/strong>\u5df1\u5728\u8bb8\u591a\u51e0\u4f55\u95ee\u9898\u4e2d\u5f97\u4ee5\u5e94\u7528\uff0c\u5305\u62ec\u5728\u89e3\u51b3Poincare\u731c\u60f3\u4e2d\u8d77\u91cd\u8981\u4f5c\u7528\u7684Ricci\u6d41\u3002
    \n3. \u6781\u5c0f\u66f2\u9762\u7684Bernstein\u95ee\u9898
    \n\u8003\u8651$\\R^n$\u4e0a\u7684\u6781\u5c0f\u66f2\u9762\u65b9\u7a0b
    \n$$
    \n\\div \\left(\\frac{\\nabla u}{\\sqrt{1+|\\nabla u|^2}}\\right)=0.
    \n$$
    \nBerstein\u57281915\u5e74\u5c31\u8bc1\u660e\u5982\u679c$n=2$, \u5219\u4efb\u4f55\u6574\u4f53\u89e3\u5fc5\u987b\u662f\u7ebf\u6027\u51fd\u6570\u3002\u5982\u679c$u$\u662f\u4e00\u4e2a\u89e3\uff0c\u5219\u5176\u56fe\u50cf
    \n$$
    \nG(u)= \\set{(x, u(x)) \\in\\R^{n+1} |x \\in\\R^n}
    \n$$
    \n\u662f\u4e00\u6781\u5c0f\u66f2\u9762\uff0c\u6216\u79f0\u201d\u6781\u5c0f\u56fe\u201d\u3002Berstein\u7684\u7ed3\u679c\u8bf4\u5e73\u9762\u4e0a\u7684\u6781\u5c0f\u56fe\u4e00\u5b9a\u662f\u5e73\u9762\u3002
    \n\u628aBerstein\u5b9a\u7406\u63a8\u5e7f\u5230$n> 2$\u7684\u52aa\u529b\u4e00\u76f4\u6ca1\u6709\u7ed3\u679c,\u76f4\u52301962\u5e74Fleming\u63d0\u51fa\u4e86\u4e00\u4e2a\u65b0\u7684\u65b9\u6cd5\u3002
    \n\u8bbe$u$\u662f\u4e00\u4e2a\u89e3\uff0c\u5219
    \n$$
    \nu_R(x)= \\frac{1}{R}u(Rx)
    \n$$
    \n\u4e5f\u662f\u89e3\u3002Fleming\u8bc1\u660e\uff0c\u5982\u679c$u$\u662f\u975e\u7ebf\u6027\u7684\uff0c$u_R$\u7684\u56fe\u5f53$R \\to\\infty$\u65f6\u5c06\u6536\u655b\u4e8e\u4e00\u4e2a$R^{n+1}$\u4e2d\u7684\u201d\u5947\u5f02\u6781\u5c0fCone\u201d$\\Sigma$\u3002\u4e8e\u662f\u95ee\u9898\u5316\u4e3a $R^{n+1}$\u4e2d\u662f\u5426\u6709\u8fd9\u6837\u7684Cone\u3002
    \n\u8fd9\u4e2a\u65b9\u6cd5\u53ef\u4ee5\u79f0\u4e3a\u201dBlow down<\/strong>\u201d, \u5373\u901a\u8fc7\u5c3a\u5ea6\u53d8\u6362\u628a\u89e3\u5728\u65e0\u7a77\u8fdc\u7684\u6e10\u8fd1\u884c\u4e3a\u62c9\u56de\u6765<\/strong>\u3002\u6cbf\u7740\u8fd9\u4e2a\u8def\u7ebf\uff0cBombieri-De Giorgi-Giusti\u57281969\u5e74\u5b8c\u5168\u89e3\u51b3\u4e86Berstein\u95ee\u9898\uff0c\u7ed3\u8bba\u662f<\/p>\n

    \u5f53$n\\leq 7$, $u$\u4e00\u5b9a\u662f\u7ebf\u6027\u7684;\u5f53$n\\geq 8$,\u5b58\u5728\u975e\u7ebf\u6027\u89e3\u3002<\/p><\/blockquote>\n

    \\section{70-80\u5e74\u4ee3}
    \n70\u5e74\u4ee3\u521d\uff0cRabinowitz, Ambrosetti\u7b49\u5f00\u59cb\u628aPalais-Smale\u53d1\u5c55\u7684\u65e0\u7a77\u7ef4\u4e34\u754c\u70b9\u7406\u8bba\u5e94\u7528\u4e8e\u5fae\u5206\u65b9\u7a0b\u95ee\u9898\uff0c\u4e3b\u8981\u662f\u534a\u7ebf\u6027\u692d\u5706\u65b9\u7a0b\u7684\u8fb9\u503c\u95ee\u9898\u89e3\u7684\u5b58\u5728\u6027\u548c\u591a\u91cd\u6027\u548cHamilton\u7cfb\u7edf\u5468\u671f\u89e3\u7684\u5b58\u5728\u6027\u3002\u8fd9\u4e9b\u5de5\u4f5c\u5bf9\u4e8e\u975e\u7ebf\u6027\u6cdb\u51fd\u5206\u6790\u4ea7\u751f\u4e86\u91cd\u8981\u5f71\u54cd\uff0c\u5728\u77ed\u77ed\u51e0\u5e74\u5185\u8fd9\u4e2a\u65b9\u5411\u7684\u8fdb\u5c55\u7a81\u98de\u731b\u8fdb\u3002\u5f53\u65f6\u7684\u9886\u8896\u662fNirenberg,\u4ed6\u57281974\u5e74\u7684\u975e\u7ebf\u6027\u6cdb\u51fd\u5206\u6790\u7684\u4e00\u4e2a\u8bb2\u4e49\u4e2d\u51e0\u4e4e\u6ca1\u6709\u63d0\u53ca\u53d8\u5206\u6cd5\uff0c\u800c\u57281981\u5e74\u7684Bulletin\u7efc\u8ff0\u6587\u7ae0\u4e2d\u7528\u4e86\u8fd1\u4e00\u534a\u7684\u7bc7\u5e45\u6765\u603b\u7ed3\u8fd9\u4e2a\u65b9\u5411\u4e0a\u7684\u6210\u679c\u3002
    \n\u4ee5\u5f20\u606d\u5e86\u4e3a\u9996\u7684\u4e2d\u56fd\u6570\u5b66\u5bb6\u572880\u5e74\u4ee3\u521d\u5f00\u59cb\u8d76\u4e0a\u6765\uff0c\u505a\u4e86\u4e00\u4e9b\u597d\u7684\u5de5\u4f5c\u3002
    \n\u73b0\u5728\u6765\u770b70\u5e74\u4ee3\u6700\u91cd\u8981\u7684\u662fRabinowitz\u5173\u4e8eHamilton\u7cfb\u7edf\u5468\u671f\u89e3\u7684\u5de5\u4f5c\u3002\u53d7\u8fd9\u4e00\u5de5\u4f5c\u7684\u5f71\u54cd\uff0cWeistein\u63d0\u51fa\u4e86Weistein\u731c\u60f3;\u8fd9\u4e2a\u731c\u60f3\u4e0eArnold\u731c\u60f3\u5bf9\u8f9b\u51e0\u4f55\u7684\u53d1\u5c55\u6709\u5f88\u5927\u63a8\u52a8\u3002
    \n\u968f\u540e\uff0c\u572880\u5e74\u4ee3\uff0c\u4e00\u7cfb\u5217\u91cd\u8981\u7684\u51e0\u4f55\u53d8\u5206\u95ee\u9898\u53d6\u5f97\u4e86\u51fa\u4eba\u610f\u6599\u7684\u8fdb\u5c55; \u540c\u65f6\uff0c\u8bb8\u591a\u91cd\u8981\u7684\u975e\u7ebf\u6027\u5206\u6790\u65b9\u6cd5\u88ab\u53d1\u73b0\u51fa\u6765\u3002<\/p>\n

      \n
    1. 2\u7ef4\u8c03\u548c\u6620\u5c04\u95ee\u9898\u3002\u572870\u5e74\u4ee3\u672b-80\u5e74\u4ee3\u521d\uff0cSacks-Uhlenback\u7814\u7a76\u4e86\u4ece\u95ed\u66f2\u9762$M$\u5230\u4efb\u610f\u7d27\u81f4\u6d41\u5f62\u7684\u8c03\u548c\u6620\u5c04\u7684\u5b58\u5728\u6027\u95ee\u9898\u3002\u7531\u4e8e $dim M=2$, \u8fd9\u4e2a\u95ee\u9898\u5177\u6709\u6240\u8c13\u201d\u5171\u5f62\u4e0d\u53d8\u6027\u201d\u3002\u4ed6\u4eec\u7684\u65b9\u6cd5\u7b80\u8ff0\u5982\u4e0b\n