$\\newcommand{\\div}{\\mathrm{div}\\,}$
\n\u6211\u4eec\u77e5\u9053\u8c03\u548c\u6620\u5c04\u7684\u65b9\u7a0b\u4e3a
\n$$
\n\\Delta u+A(u)(\\nabla u,\\nabla u)=0,
\n$$
\n\u5176\u4e2d$u:M^2\\to N$\u662f\u9ece\u66fc\u6d41\u5f62\u95f4\u7684\u6620\u5c04\u800c$A(u)$\u662f$N\\hookrightarrow \\mathbb{R}^n$\u5728$u$\u5904\u7684\u7b2c\u4e8c\u57fa\u672c\u5f62\u5f0f\u3002<\/p>\n
\u6211\u4eec\u5c06\u7528\u4e24\u79cd\u529e\u6cd5\u6765\u8bc1\u660e\u5982\u4e0b\u7684Pohozaev\u6052\u7b49\u5f0f\u3002
\n
\u73b0\u5728, \u6ce8\u610f\u5230$ru_r\\in \\Gamma(u^*TN)$, \u56e0\u6b64\u6211\u4eec\u5f97\u5230
\n\\begin{align*}
\n 0&=\\int_{B_\\rho}ru_r\\cdot \\Delta u
\n =\\int_{B_\\rho}\\div(ru_r\\cdot \\nabla u)-\\int_{B_\\rho}\\nabla(ru_r)\\cdot \\nabla u\\\\
\n &=\\int_{\\partial B_\\rho}ru_r\\cdot \\frac{\\partial u}{\\partial \\nu}-\\int_{B_\\rho}\\nabla(ru_r)\\cdot \\nabla u
\n =\\int_{\\partial B_\\rho}r \\lvert u_r \\rvert^2-\\int_{B_\\rho}\\nabla(ru_r)\\cdot \\nabla u.
\n\\end{align*}
\n\u6ce8\u610f\u5230
\n\\begin{align*}
\n \\nabla(r u_r)&=\\partial_r(ru_r)\\partial_r+r^{-2}\\partial_\\theta(ru_r)
\n =(u_r+r\\partial_r(u_r))\\partial_r+r\\cdot r^{-2}\\partial_\\theta (u_r)\\\\
\n\t &=u_r\\partial_r+r\\nabla u_r,\\\\
\n\t r \\left\\langle \\nabla u_r,\\nabla u \\right\\rangle
\n\t &=r\\left( \\left\\langle u_r, u_{rr} \\right\\rangle+r^{-2} \\left\\langle u_\\theta, u_{r\\theta} \\right\\rangle \\right)\\\\
\n\t \\frac{1}{2}r\\partial_r \\lvert \\nabla u \\rvert^2
\n\t &= \\frac{1}{2}r\\partial_r\\left( \\lvert u_r \\rvert^2+r^{-2} \\lvert u_\\theta \\rvert^2 \\right)\\\\
\n\t &=r\\left( \\left\\langle u_r, u_{rr} \\right\\rangle+r^{-2} \\left\\langle u_\\theta, u_{r\\theta} \\right\\rangle \\right)-r^{-2} \\lvert u_\\theta \\rvert^2\\\\
\n\t &=r \\left\\langle \\nabla u_r,\\nabla u \\right\\rangle-r^{-2} \\lvert u_\\theta \\rvert^2.
\n\\end{align*}
\n\u6545
\n\\begin{align*}
\n \\nabla(ru_r)\\cdot \\nabla u&= \\left\\langle u_r\\partial_r +r\\nabla u_r,\\nabla u \\right\\rangle
\n = \\lvert u_r \\rvert^2+r \\left\\langle \\nabla u_r,\\nabla u \\right\\rangle\\\\
\n\t\t\t &= \\lvert u_r \\rvert^2+r^{-2} \\lvert u_\\theta \\rvert^2+ \\frac{1}{2}r\\partial_r \\lvert \\nabla u \\rvert^2\\\\
\n\t\t\t &= \\lvert \\nabla u \\rvert^2+ \\frac{1}{2}r\\partial_r \\lvert \\nabla u \\rvert^2,
\n\\end{align*}
\n\u4ece\u800c\uff0c\u6211\u4eec\u5f97\u5230
\n\\[
\n \\int_{B_\\rho}\\nabla(ru_r)\\cdot \\nabla u
\n =\\int_{B_\\rho} \\lvert \\nabla u \\rvert^2+ \\frac{1}{2}r\\partial_r \\lvert \\nabla u \\rvert^2.
\n\\]
\n\u6ce8\u610f\u5230
\n\\begin{align*}
\n \\int_{B_\\rho}r\\partial_r \\lvert \\nabla u \\rvert^2
\n &=\\int_0^\\rho\\int_{S^1} r\\partial_r\\lvert \\nabla u \\rvert^2 rd\\theta dr
\n =\\int_0^\\rho r^2\\partial_r\\int_{S^1} \\lvert \\nabla u \\rvert^2 d\\theta dr\\\\
\n &=\\left. \\left[ r^2\\int_{S^1} \\lvert \\nabla u \\rvert^2d\\theta \\right] \\right|_{0}^\\rho-2\\int_0^\\rho r\\int_{S^1} \\lvert \\nabla u \\rvert^2 d\\theta dr\\\\
\n &=\\int_{\\partial B_\\rho}r \\lvert \\nabla u \\rvert^2 d\\sigma-2\\int_{B_\\rho} \\lvert \\nabla u \\rvert^2.
\n\\end{align*}
\n\u56e0\u6b64
\n\\begin{align*}
\n \\int_{B_\\rho}\\nabla(r u_r)\\cdot \\nabla u
\n &=\\int_{B_\\rho} \\lvert \\nabla u \\rvert^2+ \\frac{1}{2}\\int_{\\partial B_\\rho}r \\lvert \\nabla u \\rvert^2 d\\sigma-\\int_{B_\\rho} \\lvert \\nabla u \\rvert^2\\\\
\n &= \\frac{1}{2}\\int_{\\partial B_\\rho}r \\lvert \\nabla u \\rvert^2d\\sigma.
\n\\end{align*}
\n\u8fdb\u800c
\n\\begin{align*}
\n 0&=\\int_{\\partial B_\\rho}r \\lvert u_r \\rvert^2-\\int_{B_\\rho}\\nabla (r u_r)\\cdot \\nabla u\\\\
\n &=\\frac{1}{2}\\int_{\\partial B_\\rho}r\\left( \\lvert u_r \\rvert^2- r^{-2} \\lvert u_\\theta \\rvert^2 \\right),
\n\\end{align*}
\n\u5373
\n\\[
\n \\int_{\\partial B_\\rho} \\lvert u_r \\rvert^2=\\int_{\\partial B_\\rho} r^{-2} \\lvert u_\\theta \\rvert^2,
\n\\]
\n\u8fd9\u5c31\u662f\u8c03\u548c\u6620\u5c04\u7684Pohozaev\u6052\u7b49\u5f0f\u3002
\n2. \u67f1\u5750\u6807\u7cfb\u4e0b\u7684Pohozaev\u6052\u7b49\u5f0f<\/a><\/span>\n\n\u6ce8\u610f\uff0c\u82e5\u4ee4$r=e^{-t}$, \u5219$B_\\rho\\to (-\\infty,-\\ln\\rho]\\times\\theta$, $(r,\\theta)\\mapsto (t,\\theta)$. \u5bb9\u6613\u77e5\u9053$dr=-rdt$, \u4ece\u800c
\n\\[
\n ds^2=dx^2+dy^2=dr^2+r^2d\\theta^2=e^{-2t}\\left( dt^2+d\\theta^2 \\right),
\n\\]
\n\u4ee5\u53ca
\n\\begin{align*}
\n ru_r&=r \\frac{\\partial t}{\\partial r}u_t=-u_t,\\\\
\n rdr\\wedge d\\theta&=-r^2 dt\\wedge d\\theta=-e^{-2t}dt\\wedge d\\theta,\\\\
\n \\Delta u&=u_{xx}+u_{yy}= \\frac{1}{\\sqrt{g}}\\partial_\\alpha\\left( g^{\\alpha\\beta}\\sqrt{g}\\partial_\\beta u \\right)
\n =e^{2t}\\left( u_{tt}+u_{\\theta\\theta} \\right).
\n\\end{align*}
\n\u56e0\u6b64
\n\\begin{align*}
\n 0&=\\int_{B_\\rho}ru_r\\cdot \\Delta u
\n =\\int_{-\\infty}^{-\\ln\\rho}\\int_{S^1} u_t\\cdot \\left( u_{tt}+u_{\\theta\\theta} \\right) d\\theta dt\\\\
\n &=\\int_{\\ln\\rho}^\\infty\\int_{S^1} u_t\\cdot \\left( u_{tt}+u_{\\theta\\theta} \\right) d\\theta dt\\\\
\n &=\\int_{\\ln\\rho}^\\infty\\int_{S^1} \\frac{1}{2}\\partial_t \\lvert u_t \\rvert^2-u_\\theta\\cdot u_{t\\theta}+ \\int_{\\ln\\rho}^\\infty\\left. \\left[ u_t\\cdot u_\\theta \\right] \\right|_{0}^{2\\pi}dt\\\\
\n &= \\frac{1}{2}\\int_{\\ln\\rho}^\\infty\\int_0^{2\\pi} \\partial_t\\left( \\lvert u_t \\rvert^2- \\lvert u_\\theta \\rvert^2 \\right).
\n\\end{align*}
\n\u7531$\\rho$\u7684\u4efb\u610f\u6027\uff0c\u6211\u4eec\u5f97\u5230
\n\\[
\n \\partial_t\\int_{S^1}\\left( \\lvert u_t \\rvert^2- \\lvert u_\\theta \\rvert^2 \\right) d\\theta=0.
\n\\]
\n\u5728$[\\ln\\rho,+\\infty)$\u79ef\u5206\u5f97\u5230
\n\\[
\n \\int_{\\partial B_{\\rho}}\\left( \\lvert u_t \\rvert^2- \\lvert u_\\theta \\rvert^2 \\right) d\\sigma=
\n \\int_{S^1}\\left( \\lvert u_t(\\rho,\\cdot) \\rvert^2- \\lvert u_\\theta(\\rho,\\cdot) \\rvert^2 \\right) \\rho d\\theta=0,
\n\\]
\n\u8fd9\u91cc\u6211\u4eec\u4f7f\u7528\u4e86$u$\u5149\u6ed1\uff0c\u6545
\n\\[
\n \\lvert \\nabla u \\rvert^2\\leq C\\implies \\lvert u_t \\rvert^2+ \\lvert u_{\\theta} \\rvert^2\\leq C e^{-2t}=C \\rho^2,
\n\\]
\n\u5219\u8868\u660e
\n\\[
\n \\lim_{\\rho\\to 0}\\int_{S^1}\\left( \\lvert u_t(\\rho,\\cdot) \\rvert^2- \\lvert u_\\theta(\\rho,\\cdot) \\rvert^2 \\right)=0.
\n\\]<\/p>\n","protected":false},"excerpt":{"rendered":"