For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u , the shape derivative of the Dirichlet energy (with respect to u ) is shown to be well-defined and is computed in terms of u . The main difficulties in this context arise from the weak regularity of the domain and the possibly non-empty intersection of the graph of u and the transmission interface. The explicit formula for the shape derivative is then used to identify the partial differential equation solved by the minimizers of an energy functional arising in the modeling of micromechanical systems.

中文翻译：

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传输问题的狄利克雷能量的形状导数

对于位于函数 u 图形下方的截断二维圆柱体中的传输问题，狄利克雷能量的形状导数（相对于 u ）被证明是明确定义的，并根据 u 计算。这方面的主要困难来自域的弱正则性以及 u 和传输界面的图可能非空的交集。然后使用形状导数的显式公式来识别由微机械系统建模中出现的能量函数的最小值求解的偏微分方程。