\u5047\u8bbe$F=(1+\\lvert \\nabla u \\rvert^2)^{\\alpha-1}$, \u5219\u5bb9\u6613\u77e5\u9053\u03b1\u8c03\u548c\u6620\u7167\u7684\u65b9\u7a0b\u4e3a
\n$$
\n\\mathrm{div} (Fh_{ab}\\nabla u^b)-F\\Gamma_{ac}^d h_{bd}u_i^bu_i^c=0.
\n$$
\n\u5176\u4e2d, $h_{ab}$\u662f\u9776\u6d41\u5f62$N \\hookrightarrow \\mathbb{R}^K$\u7684\u5ea6\u91cf, \u800c$\\Gamma_{ab}^c$\u4e3a\u8be5\u5ea6\u91cf\u7684Christoffel\u7b26\u53f7. \u6211\u4eec\u5c06\u504f\u5bfc\u6570\u7b80\u8bb0\u4e3a$u_i^a \\mathpunct{:}=\\partial_iu^a$.<\/p>\n
\u76f4\u63a5\u5c55\u5f00\u77e5\u9053
\n$$
\nFh_{ab}\\Delta u^b+F\\nabla u^b\\cdot \\nabla u^c \\partial_ch_{ab}+\\frac{(\\alpha-1)F}{1+\\lvert \\nabla u \\rvert^2}\\nabla\\lvert \\nabla u \\rvert^2\\cdot \\nabla u^b h_{ab}-F\\Gamma_{ac}^dh_{bd} \\nabla u^b\\nabla u^c=0.
\n$$
\n\u6ce8\u610f\u5230
\n$$
\n\\Gamma_{ac}^dh_{bd}=\\frac{1}{2}\\left( \\partial_ah_{bc}+\\partial_ch_{ab}-\\partial_bh_{ac} \\right).
\n$$
\n\u6211\u4eec\u5f97\u5230
\n$$
\n\\begin{aligned}
\n&Fh_{ab}\\Delta u^b+\\frac{(\\alpha-1)F}{1+\\lvert \\nabla u \\rvert^2}h_{ab}\\nabla\\lvert \\nabla u \\rvert^2\\cdot\\nabla u^b\\\\
\n&\\qquad=\\frac{1}{2}F\\nabla u^b\\cdot\\nabla u^c\\left( \\partial_ah_{bc}+\\partial_ch_{ab}-\\partial_b h_{ac}-2\\partial_c h_{ab} \\right)\\\\
\n&\\qquad=-F\\nabla u^b\\cdot\\nabla u^c \\frac{1}{2}\\left( \\partial_bh_{ac}+\\partial_ch_{ab}-\\partial_ah_{bc} \\right)\\\\
\n&\\qquad=-F\\nabla u^b\\cdot\\nabla u^c \\Gamma_{bc}^dh_{da}.
\n\\end{aligned}
\n$$
\n\u56e0\u6b64
\n$$
\n\\Delta u^a+\\frac{(\\alpha-1)}{1+\\lvert \\nabla u \\rvert^2}\\nabla\\lvert \\nabla u \\rvert^2\\cdot\\nabla u^a=-\\Gamma_{bc}^a\\nabla u^b\\cdot\\nabla u^c.
\n$$
\n\u73b0\u5728, \u6ce8\u610f\u5230
\n$$
\n\\nabla\\lvert \\nabla u \\rvert^2\\cdot\\nabla u^a=\\nabla(h_{bc}\\nabla u^b\\nabla u^c)\\cdot\\nabla u^a
\n=\\partial_dh_{bc}(\\nabla u^d\\cdot\\nabla u^a)(\\nabla u^b\\cdot\\nabla u^c)+2h_{bc}\\nabla^2u^b(\\nabla u^c,\\nabla u^a).
\n$$
\n\u4e8e\u662f, \u6211\u4eec\u5f97\u5230\u03b1\u8c03\u548c\u6620\u7167\u7684\u65b9\u7a0b\u7684\u5f3a\u5f62\u5f0f\u4e3a
\n$$
\n\\Delta u^a+2(\\alpha-1)h_{bc}\\frac{\\nabla^2u^b(\\nabla u^c,\\nabla u^a)}{1+\\lvert \\nabla u \\rvert^2}
\n=-\\partial_dh_{bc}\\frac{\\nabla u^b\\cdot\\nabla u^c}{1+\\lvert \\nabla u \\rvert^2}(\\nabla u^a\\cdot\\nabla u^d)-\\Gamma_{bc}^a\\nabla u^b\\cdot\\nabla u^c.
\n$$
\n\u5982\u679c\u6211\u4eec\u4ee4(\u53d6\u51fa\u53d1\u6d41\u5f62\u7684\u5ea6\u91cf\u4e3a\u6b27\u6c0f\u5ea6\u91cf)
\n$$
\na_{ij}^{\\beta\\gamma}=\\delta_{ij}\\delta_{\\beta\\gamma}+\\frac{2(\\alpha-1)}{1+\\lvert \\nabla u \\rvert^2}h_{\\eta\\gamma} u_i^\\beta u_j^\\eta
\n$$
\n\u5219\u4e0a\u8ff0\u65b9\u7a0b\u53ef\u6539\u5199\u4e3a\u5982\u4e0b\u7684(\u5f53$\\alpha-1$\u5c0f\u65f6, \u4e25\u683c)\u692d\u5706\u7cfb\u7edf
\n$$
\n a_{ij}^{\\beta\\gamma}u_{ij}^\\gamma=-\\partial_\\gamma h_{\\delta\\eta}\\frac{\\nabla u^\\delta\\cdot\\nabla u^\\eta}{1+\\lvert \\nabla u\\rvert^2}\\nabla u^\\beta\\cdot\\nabla u^\\gamma-\\Gamma_{\\gamma\\delta}^\\beta\\nabla u^\\gamma\\cdot\\nabla u^\\delta.
\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"