\u6211\u4eec\u8003\u8651\u4e0a\u534a\u5706\u76d8$D$\u4e0a\u6700\u7b80\u5355\u7684Laplace\u65b9\u7a0b:
\n\\begin{equation}\\label{eq:n}
\n\\begin{cases}
\n\\Delta u=f\\in L^2(D),&x\\in D\\\\
\n\\frac{\\partial u}{\\partial \\nu}=0,&x\\in\\partial D
\n\\end{cases}
\n\\end{equation}
\n\u4e0e
\n\\begin{equation}\\label{eq:d}
\n\\begin{cases}
\n\\Delta u=f\\in L^2(D),&x\\in D\\\\
\nu=0,&x\\in\\partial D.
\n\\end{cases}
\n\\end{equation}
\n
\u4e8b\u5b9e\u4e0a, \u5bf9\\eqref{eq:n}\u53ea\u9700\u9a8c\u8bc1, \u5bf9$\\varphi\\in C^\\infty(B)$, \u6210\u7acb
\n\\begin{equation}\\label{eq:wn}
\n\\int_B\\nabla w\\nabla\\varphi+w\\varphi=0.
\n\\end{equation}
\n\u800c\u5bf9\\eqref{eq:d}\u53ea\u9700\u9a8c\u8bc1, \u5bf9$\\varphi\\in C_0^\\infty(B)$, \u6210\u7acb
\n\\begin{equation}\\label{eq:wd}
\n\\int_B\\nabla w\\nabla\\varphi+w\\varphi=0.
\n\\end{equation}
\n\u9996\u5148, \u5bf9\\eqref{eq:n}\u7684\u89e3\u6765\u9a8c\u8bc1\u3002\u5bf9$\\varphi\\in C^\\infty(B)$, \u4ee4
\n$$
\n\\varphi_e=\\frac{1}{2}(\\varphi(x)+\\varphi(x^*)),\\quad
\n\\varphi_o=\\frac{1}{2}(\\varphi(x)-\\varphi(x^*))
\n$$
\n\u5206\u522b\u4e3a\u5176\u5076\u90e8\u5206\u4e0e\u5947\u90e8\u5206\u3002\u76f4\u63a5\u8ba1\u7b97\u5f97\u5230\uff1a
\n\\begin{align*}
\n\\int_B\\nabla w\\nabla\\varphi+w\\varphi&=\\int_B\\nabla w\\nabla\\varphi_e+w\\varphi_e+
\n\\int_B\\nabla w\\nabla\\varphi_o+w\\varphi_o\\\\
\n\\int_B\\nabla w\\nabla\\varphi_e+w\\varphi_e&=\\int_D\\nabla u\\nabla\\varphi_e+u\\varphi_e+\\int_{D^-}\\nabla u(x^*)\\nabla\\varphi_e+u(x^*)\\varphi_e\\\\
\n\\int_{D^-}\\nabla u(x^*)\\nabla\\varphi_e+u(x^*)\\varphi_edx&=\\int_{D}\\nabla u\\nabla\\varphi_e(x^*)+u\\varphi_e(x^*)\\\\
\n&=\\int_D\\nabla u\\nabla\\varphi_e+u\\varphi_e\\\\
\n\\int_B\\nabla w\\nabla\\varphi_o+w\\varphi_o&=\\int_D\\nabla w\\nabla\\varphi_o+w\\varphi_o+\\int_{D^-}\\nabla w\\nabla\\varphi_o+w\\varphi_o\\\\
\n&=\\int_D\\nabla u\\nabla\\varphi_o+u\\varphi_o+\\int_{D^-}\\nabla u(x^*)\\nabla\\varphi_o+u(x^*)\\varphi_o\\\\
\n\\int_{D^-}\\nabla u(x^*)\\nabla\\varphi_o+u(x^*)\\varphi_o&=\\int_D\\nabla u\\nabla\\varphi_o(x^*)+u\\varphi_o(x^*)\\\\
\n&=-\\int_D\\nabla u\\nabla\\varphi_o+u\\varphi_o
\n\\end{align*}
\n\u7531\u4e8e$\\varphi\\in C^\\infty(B)$\u6545$\\varphi_e\\in C^\\infty(D)$. \u800c$u$\u662f\\eqref{eq:n}\u5728$D$\u4e0a\u7684$W^{1,2}$\u5f31\u89e3, \u4e8e\u662f
\n$$
\n\\int_D\\nabla u\\nabla\\varphi_e+u\\varphi_e=0.
\n$$
\n\u8fd9\u6837, \u6211\u4eec\u5c31\u8bc1\u660e\u4e86
\n$$
\n\\int_B\\nabla w\\nabla\\varphi+w\\varphi=0,\\quad\\varphi\\in C^\\infty(B).
\n$$<\/p>\n
\u5bf9\\eqref{eq:d}\u7684\u89e3\u7684\u9a8c\u8bc1\u662f\u7c7b\u4f3c\u7684. \u53ea\u9700\u6ce8\u610f\u5230, \u5bf9$\\varphi=\\varphi_e+\\varphi_o\\in C_0^\\infty(B)$, \u548c\u524d\u9762\u5bf9Neumann\u60c5\u5f62\u7684\u9a8c\u8bc1\u4e00\u6837, \u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5f97\u5230(\u6ce8\u610f\u6b64\u65f6$w$\u662f\u5947\u5ef6\u62d3),
\n\\begin{align*}
\n\\int_B\\nabla w\\nabla\\varphi+w\\varphi&=\\int_B\\nabla w\\nabla\\varphi_e+w\\varphi_e+
\n\\int_B\\nabla w\\nabla\\varphi_o+w\\varphi_o\\\\
\n\\int_B\\nabla w\\nabla\\varphi_e+w\\varphi_e&=\\int_D\\nabla u\\nabla\\varphi_e+u\\varphi_e\\color{red}{-}\\int_{D^-}\\nabla u(x^*)\\nabla\\varphi_e+u(x^*)\\varphi_e\\\\
\n&=0\\\\
\n\\int_B\\nabla w\\nabla\\varphi_o+w\\varphi_o&=\\int_D\\nabla w\\nabla\\varphi_o+w\\varphi_o+\\int_{D^-}\\nabla w\\nabla\\varphi_o+w\\varphi_o\\\\
\n&=\\int_D\\nabla u\\nabla\\varphi_o+u\\varphi_o\\color{red}{-}\\int_{D^-}\\nabla u(x^*)\\nabla\\varphi_o+u(x^*)\\varphi_o\\\\
\n&=2\\int_D\\nabla u\\nabla\\varphi_o+u\\varphi_o.
\n\\end{align*}
\n\u7531\u4e8e$\\varphi\\in C_0^\\infty(B)$, \u6211\u4eec\u77e5\u9053$\\varphi_o\\in C_0^\\infty(B)$\u4e14$\\varphi_o|_{\\partial^0D}=0$. \u8fd9\u91cc, $\\partial^0D=\\{x=(x^1,x^2)\\in D:x_2=0\\}$, $\\partial^+D=\\{x\\in\\partial D:x^2>0\\}$. \u56e0\u6b64, \u5b9e\u9645\u4e0a$\\varphi_o\\in C_0^\\infty(D)$. \u6545\u7531$u$\u662f\\eqref{eq:d}\u5728$D$\u4e0a\u7684\u5f31\u89e3\u77e5\u9053
\n$$
\n\\int_D\\nabla u\\nabla\\varphi_o+u\\varphi_o=0.
\n$$
\n\u8fd9\u6837, \u6211\u4eec\u5c31\u8bc1\u660e\u4e86
\n$$
\n\\int_B\\nabla w\\nabla\\varphi+w\\varphi=0,\\quad\\varphi\\in C_0^\\infty(B).
\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"