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![\"cutoff-phi.png\"](\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2017\/06\/cutoff-phi.png\")
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![\"cutoff-dphi.png\"](\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2017\/06\/cutoff-dphi.png\")
\u5728\u5b9e\u9645\u5e94\u7528\u8fc7\u7a0b\u4e2d, \u6211\u4eec\u9700\u8981\u6784\u9020\u6ee1\u8db3\u5982\u4e0b\u6761\u4ef6\u7684\u622a\u65ad\u51fd\u6570:
\n$$
\n 0\\leq\\phi\\leq 1,\\quad\\phi|_{B_{1}}\\equiv1,\\quad \\mathrm{supp}\\phi\\subset B_2,\\quad\\lvert \\nabla\\phi\\rvert\/\\phi\\leq 2.
\n$$
\n\u4e0a\u8bc9\u622a\u65ad\u51fd\u6570\u53ef\u4ee5\u5177\u4f53\u7684\u6784\u9020\u4e3a:
\n\\begin{align*}
\nf(t)&=\\begin{cases}
\ne^{-1\/t},&t>0\\\\
\n0,&t\\leq0
\n\\end{cases}\\\\
\ng(t)&=\\frac{f(2-t)}{f(t-1)+f(2-t)}\\\\
\n\\phi(t)&=g(|t|)
\n\\end{align*}
\n\u53ef\u4ee5\u9a8c\u8bc1
\n\\begin{gather*}
\n\\phi\\in C^\\infty,\\quad
\n\\phi|_{B_1}\\equiv 1,\\quad
\n\\phi|_{B_2^c}\\equiv 0,\\quad
\n0\\leq\\phi\\leq 1\\\\
\n\\phi’\/\\phi=\\begin{cases}
\n\\frac{-(5-6|t|+2t^2)\\exp\\left(\\frac{1}{2-3|t|+t^2}\\right)\\mathrm{Sign}(t)}{\\left[(2-3|t|+t^2)\\left(\\exp\\left(\\frac{1}{1-|t|}\\right)+\\exp\\left(\\frac{1}{|t|-2}\\right)\\right)\\right]^2},&1<|t|<2\\\\\n0,&\\text{otherwise}\n\\end{cases}\n\\end{gather*}<\/p>\n
\u5b83\u4eec\u7684\u56fe\u50cf\u5206\u522b\u5982\u4e0b
\n
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