The Extreme Problem For Orlicz And L_q Torsional Rigidity And Their Properties

In this thesis,based on the concept of Orlicz L_qmixed torsional rigidity,we introduce Orlicz and L_qgeominimal torsional rigidity and Orlicz and L_q mixed geominimal torsional rigidity,and prove some good properties of them,such as their invariance under orthogonal matrix,corresponding isoperimetric type inequalities,cyclic type inequalities and the Aleksandrov-Fenchel inequality.The first chapter mainly introduces the research status and progress of torsional rigidity.The second chapter mainly introduces some basic symbols and concepts,the definition of torsional rigidity,and the related inequalities required in the demonstration process.In Chapter 3,we establish the homogeneous Orlicz L_q mixed torsional rigidity.Based on this,we define Orlicz geominimal torsional rigidity.According to the definition of Orlicz geominimal torsional rigidity,its invariance under orthogonal matrix and the corresponding isoperimetric type inequality are proved.Based on the definition of Orlicz L_q mixed torsional rigidity in Chapter 3,in Chapter 4,we introduce L_q geominimal torsional rigidity and study its related properties,that is,the isoperimetric type inequality and cyclic type inequality of L_q geominimal torsional rigidity,and the invariance of L_q geominimal torsional rigidity under orthogonal matrix.Based on the research on Orlicz and L_q geominimal torsional rigidity and their related properties in Chapter 3 and 4,in Chapter 5,the L_q geominimal torsional rigidity related to a single convex body is extended to the L_q mixed geominimal torsional rigidity related to multiple convex bodies,and the invariance of L_q mixed geominimal torsional rigidity related to multiple convex bodies under orthogonal matrix is proved,and its corresponding isoperimetric type inequality,cyclic type inequality and Aleksandrov-Fenchel type inequality are established.