{"id":366,"date":"2017-01-19T07:02:52","date_gmt":"2017-01-19T07:02:52","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=366"},"modified":"2017-01-19T07:02:52","modified_gmt":"2017-01-19T07:02:52","slug":"pianweifenfangchengqimoshiti","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=366","title":{"rendered":"\u504f\u5fae\u5206\u65b9\u7a0b\u671f\u672b\u8bd5\u9898"},"content":{"rendered":"
  1. \u82e5$\\Omega=(0,2)\\times(0,2)\\subset\\mathbb{R}^2$, $u(x_1,x_2)$\u4e3a\u5982\u4e0b\u65b9\u7a0b\u7684\u89e3
    1. (8′)\u6c42
      \n\\begin{cases}
      \n \\Delta u+\\lambda u=0, &x\\in\\Omega\\\\
      \n u|_{\\partial\\Omega}=0
      \n\\end{cases}
      \n\u7684\u7b2c\u4e00\u3001\u7b2c\u4e8c\u7279\u5f81\u503c\u4ee5\u53ca\u76f8\u5e94\u7684\u7279\u5f81\u51fd\u6570.<\/li>
    2. (7′)\u5bf9\u54ea\u4e9b$a$, \u65b9\u7a0b
      \n\\begin{cases}
      \n \\Delta u+\\frac{\\pi^2}{2}u=x_1-a,&x\\in\\Omega\\\\
      \n u|_{\\partial\\Omega}=0
      \n\\end{cases}
      \n\u81f3\u5c11\u6709\u4e00\u89e3, \u8bf4\u660e\u7406\u7531.<\/li><\/ol><\/li>
    3. (10′)\u8bbe$u(x,y)$\u4e3a\u65b9\u7a0b
      \n\\begin{cases}
      \n\\Delta u=x+y,&(x,y)\\in B_1(0)\\subset\\mathbb{R}^2,\\\\
      \nu|_{\\partial B_1(0)}=0
      \n\\end{cases}
      \n\u7684\u89e3, \u6c42$u(0,0)$.<\/li>
    4. (15′)\u8bbe$\\Omega=\\set{(x_1,x_2)|1<|x|<2}$, \u6c42\u6781\u5c0f\n\\[\n\\inf_{w-(|x|^2-1)\\in H_0^1(\\Omega)}\\int_{\\Omega}(|\\nabla w|^2-2w)dx.\n\\]\n<\/li>
    5. \u8bbe$x=(x_1,x_2)\\in B_1(0)\\subset\\mathbb{R}^2$, \n
        \n
      1. (5')\u82e5$u\\in C^\\infty(\\overline{B_1(0)})$\u4e3a\u65b9\u7a0b$\\Delta u=0$, $x\\in B_1(0)$\u7684\u89e3, \u6c42\u8bc1\n\\[\n\\sup_{B_1(0)}|\\nabla u|\\leq\\sup_{\\partial B_1(0)}|\\nabla u|.\n\\]\n<\/li>
      2. (10')\u82e5$u\\in C^\\infty(\\overline{B_1(0)})$\u4e3a\u65b9\u7a0b$\\Delta u+u=0$, $x\\in B_1(0)$\u7684\u89e3, \u6c42\u8bc1: \u5b58\u5728$C>0$(\u4f8b\u5982\u53ef\u53d6$C=100$), \u4f7f\u5f97
        \n\\[
        \n\\sup_{B_1(0)}|\\nabla u|\\leq C\\left(\\sup_{\\partial B_1(0)}|\\nabla u|+\\sup_{B_1(0)}|u|\\right).
        \n\\]<\/li>
      3. (15′)\u82e5$u>0$, \u4e14$u\\in C^\\infty(\\overline{B_1(0)})$\u4e3a\u65b9\u7a0b$\\Delta u+u=0$, $x\\in B_1(0)$\u7684\u89e3, \u6c42\u8bc1\u5b58\u5728$C>0$(\u4f8b\u5982\u53d6$C=3^{10}$), \u4f7f\u5f97
        \n\\[
        \n\\sup_{B_{1\/2}(0)}u\\leq C\\inf_{B_{1\/2}(0)}u.
        \n\\]<\/li><\/ol><\/li>
      4. (15′)\u8bbe$u(x,t)$\u4e3a\u65b9\u7a0b
        \n\\[
        \n\\begin{cases}
        \nu_t=u_{xx},&0< x< 1, t >0\\\\
        \nu|_{x=0}=u|_{x=1}=0,\\\\
        \nu|_{t=0}=x(1-x),
        \n\\end{cases}
        \n\\]
        \n\u7684\u89e3, \u6c42\u8bc1
        \n\\[
        \n\\int_0^1u^2(x,t)dx\\leq\\frac{e^{-2\\pi^2t}}{30}.
        \n\\]<\/li>
      5. (15′)\u8bbe$u(x,t)$\u4e3a\u65b9\u7a0b
        \n\\begin{cases}
        \nu_{tt}=u_{xx},&0< x< 1,\\,t >0\\\\
        \nu|_{x=0}=u|_{x=1}=0,\\\\
        \nu|_{t=0}=\\sin(\\pi x),\\\\
        \nu_{t}|_{t=0}=x,
        \n\\end{cases}
        \n\u7684\u89e3, \u4ee4
        \n\\[
        \nE(t)=\\int_0^1(u_t^2+u_x^2)dx,
        \n\\]
        \n\u6c42$E(3)$\u7684\u503c.<\/li>
      6. (10′)\u8bbe$U\\subset\\mathbb{R}^n$\u4e3a\u6709\u754c\u5f00\u96c6, \u5bf9\u67d0\u4e2a\u56fa\u5b9a\u7684$T>0$, $U_T=U\\times(0,T]$,
        \n$$
        \nLu:=-\\sum_{i,j=1}^na^{ij}(x,t)u_{ij}+b^i(x,t)u_i+c(x,t)u,
        \n$$
        \n\u5176\u4e2d$a^{ij}(x,t)$, $b^i$, $c\\in C(\\overline{U_T})$, \u4e14\u5bf9\u6240\u6709\u7684$(x,t)\\in U_T$\u4ee5\u53ca\u6240\u6709\u7684$\\xi\\in\\mathbb{R}^n$\u6709$a^{ij}=a^{ji}$, $a^{ij}(x,t)\\xi_i\\xi_j\\geq\\alpha_0|\\xi|^2$, $\\alpha_0>0$. \u8bbe$u\\in C_1^2(U_t)\\cap C(\\overline{U_T})$\u4e14$c\\geq0$\u5728$U_T$\u4e0a\u6210\u7acb. \u82e5$u_t+Lu\\leq0$\u5728$U_T$\u4e0a\u6210\u7acb, \u4e14$u$\u5728$(x_0,t_0)\\in U_T$\u4e0a\u8fbe\u5230\u5b83\u5728$\\overline{U_T}$\u4e2d\u7684\u6700\u5927\u503c$0$, \u5219$u\\equiv0$\u5728$U_{t_0}$\u4e0a\u6210\u7acb.<\/li>
      7. \u8bbe$\\Omega\\subset\\mathbb{R}^n$\u4e3a\u6709\u754c\u5149\u6ed1\u533a\u57df, \u4e14\u5bf9\u4efb\u610f\u7684$x\\in\\overline{\\Omega}$, \u4ee5\u53ca\u4efb\u4f55$\\xi\\in\\mathbb{R}^n\\setminus\\set{0}$\u6210\u7acb$a^{ij}(x)\\in C^\\infty(\\bar\\Omega)$, $a^{ij}(x)=a^{ji}(x)$, $a^{ij}(x)\\xi_i\\xi_j\\geq\\alpha_0|\\xi|^2$, $\\alpha_0>0$. \u82e5$u\\in C^\\infty(\\overline{Q_T})$, $Q_T=\\Omega\\times(0,T)$, \u662f\u65b9\u7a0b
        \n\\[
        \n\\begin{cases}
        \nu_t-\\sum_{i,j=1}^n(a^{ij}(x)u_i)_j=f(x,t)\\in C^\\infty(\\overline{Q_T}),&x\\in Q_T,\\\\
        \nu|_{\\partial\\Omega}=0,\\\\
        \nu|_{t=0}=g\\in H_0^1(\\Omega),
        \n\\end{cases}
        \n\\]
        \n\u7684\u89e3, \u6c42\u8bc1: \u5bf9\u4efb\u4f55$0\\leq t\\leq T$, \u6709\u5982\u4e0b\u4f30\u8ba1\u6210\u7acb
        1. (15′)
          \n\\[
          \n\\max_{0\\leq t\\leq T}\\|u(t)\\|_{L^3(\\Omega)}+\\|u\\|_{L^2(0,T;H_0^1(\\Omega))}\\leq c\\left(\\|f\\|_{L^2(0,T;L^2(\\Omega))}+\\|g\\|_{L^2(\\Omega)}\\right);
          \n\\]<\/li>
        2. (15′)
          \n\\[
          \n\\sup_{0\\leq t\\leq T}\\|u(t)\\|_{H_0^1(\\Omega)}+\\|u\\|_{L^2(0,T;H^2(\\Omega))}\\leq c\\left(\\|f\\|_{L^2(0,T;L^2(\\Omega))}+\\|g\\|_{H_0^1(\\Omega)}\\right).
          \n\\]<\/li><\/ol><\/li>
        3. (20′)\u8bbe$\\Omega\\subset\\mathbb{R}^n$\u4e3a\u6709\u754c\u5149\u6ed1\u533a\u57df, $Q_T=\\Omega\\times(0,T)$,
          \n\\[
          \nLu:=-\\sum_{i,j=1}^n(a^{ij}(x)u_i)_j+b^i(x,t)u_i+c(x,t)u,
          \n\\]
          \n\u5176\u4e2d$a^{ij}(x)$, $b^i$, $c\\in C^\\infty(\\overline{Q_T})$, \u4e14\u5bf9\u4efb\u4f55\u7684$x\\in\\bar\\Omega$, \u4ee5\u53ca\u4efb\u4f55\u7684$\\xi\\in\\mathbb{R}^n\\setminus\\set{0}$, \u6709$a^{ij}(x)=a^{ji}(x)$, $a^{ij}(x)\\xi_i\\xi_j\\geq\\alpha_0|\\xi|^2$, $\\alpha_0>0$. \u82e5$u\\in C^\\infty(\\overline{Q_T})$\u4e3a\u65b9\u7a0b
          \n\\[
          \n\\begin{cases}
          \nu_{tt}+Lu=f(x,t)\\in L^2(0,T;L^2(\\Omega)),&x\\in Q_T\\\\
          \nu|_{\\partial \\Omega}=0,\\\\
          \nu|_{t=0}=g\\in H_0^1(\\Omega),\\\\
          \nu_t|_{t=0}=h\\in L^2(\\Omega),
          \n\\end{cases}
          \n\\]
          \n\u7684\u89e3, \u4ee4
          \n\\[
          \nE(t)=\\int_{\\Omega}(u^2+u_t^2+|\\nabla_xu|^2)dx,
          \n\\]
          \n\u6c42\u8bc1:
          \n\\[
          \nE(t)\\leq c\\left(\\int_0^T\\int_\\Omega f^2(x,t)dxdt+\\|g\\|^2_{H_0^1(\\Omega)}+\\|h\\|_{L^2(\\Omega)}^2\\right),
          \n\\]
          \n\u5176\u4e2d$c>0$\u4e14\u53ea\u4e0e$a^{ij}$, $b^i(x,t)$, $c(x,t)$\u6709\u5173.<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

          \u82e5$\\Omega=(0,2)\\times(0,2)\\subset\\mathbb{R}^2$, $u(x_1,x… \u7ee7\u7eed\u9605\u8bfb\u504f\u5fae\u5206\u65b9\u7a0b\u671f\u672b\u8bd5\u9898<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[67,64],"class_list":["post-366","post","type-post","status-publish","format-standard","hentry","category-math","tag-pianweifenfangcheng","tag-shiti","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/366","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=366"}],"version-history":[{"count":11,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/366\/revisions"}],"predecessor-version":[{"id":377,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/366\/revisions\/377"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=366"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=366"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=366"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}