In this paper, we study the steady-state distribution of heat on long pipes in \({\mathbb {R}}^3\) heated along some regions of their surfaces. In particular, we prove that, if the pipe \(P=\{(x,y,z):\,x^2+y^2<1\}\) is heated along its surface belt \(B(a)=\{(x,y,z):\,x^2+y^2=1,-a<z<a\}\), \(a>0\), then the temperature in its cross-sections \(D_c=\{(x,y,z)\in P:\, z=c\}\) is increasing in the radial direction for all c in the interval \([-a, a]\).

中文翻译：

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加热长管

在本文中，我们研究了沿其表面某些区域受热的\（{{mathbb {R}} ^ 3 \）中长管上热量的稳态分布。特别是，我们证明，如果管道\（P = \ {（x，y，z）：\，x ^ 2 + y ^ 2 <1 \} \）沿其表面带\（B（a ）= \ {（x，y，z）：\，x ^ 2 + y ^ 2 = 1，-a <z <a \} \），\（a> 0 \），则其交叉点处的温度对于区间c （[-a，a] \）中的所有c，截面\（D_c = \ {{x，y，z）\ in P：\，z = c \} \）沿径向方向增加。