The Relevant Properties Of The Torsional Rigidity Under The Orlicz Addition

In this paper,we consider the optimization problems associated with the nonhomogeneous and homogeneous Orlicz mixed torsional rigidities by investigating the properties of the corresponding mixed torsional rigidity.The existence and continuity of the solutions of these problems are proved,and the corresponding isoperimetric type inequalities are established.The first chapter mainly introduces the research background of the torsional rigidity.In the second chapter introduces the definition of torsional rigidity.In the third chapter,we first give the definitions of nonhomogeneous and homogeneous Orlicz mixed torsional rigidities,and then according to the definitions,we study the continuity,uniformly boundedness,rigidity invariance and other good properties of nonhomogeneous and homogeneous Orlicz mixed torsional rigidities,respectively.In the fourth chapter,we introduce definitions of nonhomogeneous and homogeneous Orlicz geominimal torsional rigidities,and study the related extreme value problems,at the same time prove that the existence of nonhomogeneous and homogeneous Orlicz mixed torsional rigidity Petty body,further discuss the related properties of the Petty body,such as continuity and boundedness,make use of these properties we establish this Petty body isoperimetric type inequality.In particular,a counter-example is given to show that the the extreme value problem of nonhomogeneous Orlicz mixed torsional rigidity has no solution in general,that is,the solution of the extreme value problem does not exist in general.In the fourth chapter,we also study Orlicz mixed torsional rigidities composed of convex bodies.By using the definition of this Orlicz mixed torsional rigidities,we prove the continuity of Orlicz mixed torsional rigidities composed of convex bodies and the existence of Orlicz mixed torsional rigidities Petty body composed of convex bodies.