We consider shape functionals of the form \(F_q(\Omega )=P(\Omega )T^q(\Omega )\) on the class of open sets of prescribed Lebesgue measure. Here \(q>0\) is fixed, \(P(\Omega )\) denotes the perimeter of \(\Omega \) and \(T(\Omega )\) is the torsional rigidity of \(\Omega \). The minimization and maximization of \(F_q(\Omega )\) is considered on various classes of admissible domains \(\Omega \): in the class \(\mathcal {A}_{all}\) of all domains, in the class \(\mathcal {A}_{convex}\) of convex domains, and in the class \(\mathcal {A}_{thin}\) of thin domains.

中文翻译：

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涉及周长和扭转刚度的一些不等式

我们考虑形式为（（F_q（\ Omega）= P（\ Omega）T ^ q（\ Omega）\）的形状函数。这里\（Q> 0 \）是固定的，\（P（\欧米茄）\）表示的周边\（\欧米茄\）和\（T（\欧米茄）\）被扭转的刚度\（\欧米茄\ ）。在所有类别的允许域\（\ Omega \）中考虑\（F_q（\ Omega）\）的最小化和最大化：在所有域的\（\ mathcal {A} _ {all} \）类中，在类\（\ mathcal {A} _ {凸} \）的凸结构域，和在类\（\ mathcal {A} _ {thin} \）的薄域。