\u56de\u5fc6, $\\alpha$-\u6768\u7c73\u5c14\u65af–\u5e0c\u683c\u65af\u573a\u7684\u65b9\u7a0b\u4e3a\u65b9\u7a0b\u7ec4
\n$$
\n\\Delta a_j^\\beta=2\\left( -a_i^\\gamma \\partial_ia_j^\\delta g_{\\gamma\\delta}^\\beta+\\Upsilon h_{ab}(u)u_{|j}^a\\lambda_\\beta^b(u)+F_{ij}^\\gamma a_i^\\delta g_{\\beta\\delta}^\\gamma \\right),
\n$$
\n\u4e0e
\n$$
\n\\mathrm{div}(\\Upsilon h_{ab}u_{|i}^a)=\\Upsilon h_{ad}(u)u_{|i}^a\\left( \\Gamma_{bc}^d(u)\\partial_iu^c+A_{bi}^d(u) \\right)+\\mu(u)\\cdot\\left[ \\nabla_{\\partial_{f^b}}\\mu \\right](u),
\n$$
\n\u5176\u4e2d$\\Upsilon=\\alpha(1+\\lvert \\nabla_Au \\rvert^2)^{\\alpha-1}=\\alpha\\left(1+h_{ab}(u)(\\partial_iu^a+a_i^\\beta\\lambda_\\beta^a(u))(\\partial_iu^b+a_i^\\gamma\\lambda_\\gamma^b(u))\\right)^{\\alpha-1}$, $u_{|i}^a=\\partial_iu^a+a_i^\\beta \\lambda_\\beta^a(u)$, $F_{ij}^\\gamma=(\\partial_ia_j^\\gamma-\\partial_ja_i^\\gamma)+2a_i^\\beta a_j^\\delta g_{\\beta\\delta}^\\gamma$, $A_{bi}^d(u)=a_i^\\beta\\left(\\partial_{f^b}\\lambda_\\beta^d(u)+\\lambda_\\beta^c(u)\\Gamma_{bc}^d(u)\\right)$, \u800c$h$, $\\Gamma$, $\\lambda$, $\\mu$ \u662f $u$\u7684\u5149\u6ed1\u51fd\u6570. \u6309\u7167\u5b9a\u4e49 $[v_\\beta,v_\\gamma]=g_{\\beta\\gamma}^\\delta v_\\delta$, $\\left\\{ g_{\\beta\\gamma}^\\delta \\right\\}$ \u79f0\u4e3a\u674e\u4ee3\u6570\u7684\u7ed3\u6784\u5e38\u6570.
\n<\/p>\n
\u73b0\u5728, \u6ce8\u610f\u5230 \u5b8c\u5168\u7c7b\u4f3c\u5730, \u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u5173\u4e8e\u8054\u7edc$a$\u7684\u65b9\u7a0b.
\n$$
\n\\mathrm{div}(\\Upsilon h_{ab}u_{|i}^a)=\\Upsilon h_{ab}(u)\\partial_i u_{|i}^a+\\Upsilon\\partial_ch_{ab}u_{|i}^a\\partial_iu^c+\\partial_i\\Upsilon h_{ab}(u)u_{|i}^a
\n\\mathpunct{:}=\\o{1}+\\o{2}+\\o{3}.
\n$$
\n\u4e3a\u65b9\u4fbf\u8d77\u89c1, \u53c8\u4ee4
\n$$
\n \\begin{aligned}
\n \\o{4}&\\mathpunct{:}=\\Upsilon h_{ad}u_{|i}^a\\left( \\Gamma_{bc}^d\\partial_iu^c+A_{bi}^d \\right),\\\\
\n \\o{5}&\\mathpunct{:}=\\mu\\cdot\\nabla_{\\partial_{f^b}}\\mu.
\n \\end{aligned}
\n$$
\n\u5219\u76f4\u63a5\u8ba1\u7b97\u5f97\u5230
\n$$
\n\\begin{aligned}
\n \\o{1}&=\\Upsilon h_{ab}\\Delta u^a+\\Upsilon h_{ab}\\partial_i(a_i^\\beta\\lambda_\\beta^a),\\\\
\n \\o{2}&=\\Upsilon \\partial_ch_{ab}\\nabla u^a\\cdot\\nabla u^c+\\Upsilon\\partial_ch_{ab}\\partial_iu^ca_i^\\beta\\lambda_\\beta^a,\\\\
\n \\o{3}&=\\frac{2(\\alpha-1)\\Upsilon}{1+\\lvert \\nabla_Au \\rvert^2}h_{ab}\\left( \\partial_ju^a+a_j^\\beta\\lambda_\\beta^a \\right)
\n h_{cd}\\left( \\partial_iu^c+a_i^\\beta\\lambda_\\beta^c \\right)
\n \\left( \\partial_{ij}u^d+\\partial_i(a_j^\\beta \\lambda_\\beta^d(u)) \\right)\\\\
\n &\\qquad+\\frac{(\\alpha-1)\\Upsilon}{1+\\lvert \\nabla_Au \\rvert^2}h_{ab}u_{|j}^a\\partial_eh_{cd}\\partial_ju^eu_{|i}^cu_{|i}^d,\\\\
\n \\o{4}&=\\Upsilon h_{ad}\\Gamma_{bc}^d\\partial_iu^a\\partial_iu^c+\\Upsilon h_{ad}a_i^{\\beta}\\lambda_\\beta^a\\Gamma_{bc}^d\\partial_iu^c
\n +\\Upsilon h_{ad}(\\partial_iu^a+a_i^\\beta\\lambda_\\beta^a)A_{bi}^d.
\n\\end{aligned}
\n$$
\n\u6ce8\u610f\u5230$h$,$\\Gamma$, $\\lambda$, $\\mu$, \u4f5c\u4e3a$u$\u7684\u51fd\u6570\u5b83\u4eec\u672c\u8eab\u53ca\u5176\u5bfc\u6570\u90fd\u662f\u6709\u754c\u7684(\u56e0\u4e3a\u9776\u6d41\u5f62\u662f\u7d27\u81f4\u7684), \u5982\u4e0b, \u6211\u4eec\u5c06\u7528$\\#$\u6765\u8868\u793a\u7cfb\u6570\u4e3a\u6709\u754c\u51fd\u6570\u7684\u4e58\u79ef[1<\/a>]<\/sup>u^d=\\nabla u\\#\\nabla u+\\nabla a+a\\#\\nabla u+a\\#a+\\mu\\cdot\\nabla\\mu.
\n$$<\/p>\n
\n$$
\n\\Delta a=a\\#\\nabla a+\\Upsilon\\#(\\nabla u+a)+(\\nabla a+a\\#a)\\#a.
\n$$
\n\u5373
\n$$
\n\\Delta a=a\\#\\nabla a+\\Upsilon\\#(\\nabla u+a)+a\\#a\\#a
\n$$
\n\u8fd9\u91cc, $\\#$\u8fd8\u4f9d\u8d56\u4e8e\u674e\u4ee3\u6570\u7684\u7ed3\u6784\u5e38\u6570\u7684$L^\\infty$\u6a21.<\/p>\n
\n\t\n\t\t
\n$$
\n\\begin{aligned}
\n \\o{1}&=\\Upsilon h_{ab}\\Delta u^a+\\Upsilon h_{ab}\\left( \\nabla a+a\\#\\nabla u \\right),\\\\
\n \\o{2}&=\\Upsilon \\partial_ch_{ab}\\nabla u^a\\cdot \\nabla u^c+\\Upsilon a\\#\\nabla u,\\\\
\n \\o{3}&=2(\\alpha-1)\\Upsilon h_{ab} \\frac{h_{cd}u_{|i}^au_{|j}^c}{1+h_{cd}u_{|i}^cu_{|i}^d}\\left( \\partial_{ij}u^d+\\nabla a+a\\#\\nabla u \\right)\\\\
\n &\\quad+(\\alpha-1)\\Upsilon h_{ab}\\frac{\\partial_eh_{cd}u_{|i}^cu_{|i}^d}{1+\\lvert \\nabla_Au \\rvert^2}u_{|j}^a\\partial_ju^e\\\\
\n &=2(\\alpha-1)\\Upsilon h_{ab} \\frac{h_{cd}u_{|i}^au_{|j}^c}{1+h_{cd}u_{|i}^cu_{|i}^d}\\partial_{ij}u^d+(\\alpha-1)\\Upsilon h_{ab}\\frac{\\partial_eh_{cd}u_{|i}^cu_{|i}^d}{1+\\lvert \\nabla_Au \\rvert^2}u_{|j}^a\\partial_ju^e+\\Upsilon h_{ab}\\left( \\nabla a+a\\#\\nabla u +\\nabla u\\#\\nabla u\\right)\\\\
\n \\o{4}&=\\Upsilon h_{ad}\\Gamma_{bc}^d\\nabla u^a\\cdot\\nabla u^c+\\Upsilon h_{ad}(a\\#\\nabla u+(\\nabla u+a)\\# a),\\\\
\n \\o{5}&=\\mu\\cdot\\nabla\\mu
\n\\end{aligned}
\n$$
\n\u5173\u4e8e$u$\u7684\u65b9\u7a0b\u53ef\u4ee5\u5316\u4e3a
\n$$
\n\\Delta u^a+2(\\alpha-1) \\frac{h_{cd}u_{|i}^au_{|j}^c}{1+h_{cd}u_{|i}^c u_{|i}^d}\\partial_{ij}u^d
\n=(\\alpha-1)\\frac{\\partial_eh_{cd}u_{|i}^cu_{|i}^d}{1+\\lvert \\nabla_Au \\rvert^2}u_{|j}^a\\partial_ju^e-\\Gamma_{bc}^a\\nabla u^b\\cdot \\nabla u^c+\\nabla a+a\\#\\nabla u+a\\#a+\\mu\\cdot\\nabla\\mu.
\n$$
\n\u5373
\n$$
\n\\Delta u^a+(\\alpha-1) \\frac{2h_{cd}u_{|i}^au_{|j}^c}{1+h_{cd}u_{|i}^c u_{|i}^d}\\partial_{ij<\/li>\n\t<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"