Applied Mathematics and Optimization ( IF 1.6 ) Pub Date : 2020-10-21 , DOI: 10.1007/s00245-020-09727-7 Luca Briani , Giuseppe Buttazzo , Francesca Prinari
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:
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。