The Uniqueness Of L_p-Minkowski Problem For Q-Torsional Rigidity

Convex geometry analysis is widely used in many fields,such as mathematic eco-nomics,partial differential equations,probability theory,computer pattern recognition,CT scanning and nuclear magnetic resonance in medicine.Convex geometry analysis mainly studies geometric structures and invariants of geometric bodies(which mainly include convex bodies and stars bodies)by using the methods of geometry and anal-ysis,with Brunn-Minkowski theory as its core content.Minkowski problem plays an important role in Brunn-Minkowski theory.It is a characteristic problem of geometric measures generated by convex bodies.Minkowski problem is always one of the focus problems in convex geometry analysis,integral geometry,differential geometry and partial differential equations.It mainly involves the existence,uniqueness,continuity and regularity of solutions in partial differential equations.In this paper,we principal-ly use the basic knowledge and methods ofL_p-Brunn-Minkowski theory to study the uniqueness ofL_p-Minkowski problem ofq-torsional rigidity.Next,we will introduce the major content of this paper,including the following parts:In Chapter 1,firstly,we introduce the research background and research status of this paper,which mainly include the background ofL_p-Brunn-Minkowski theory and its related theories;secondly,the structure and main results of this paper is given;finally,the symbolic description of our works is given.In Chapter 2,we first introduce some basic concepts and facts of convex bodies in convex geometry analysis,such as support function,Minkowski addition,L_p-Minkowski linear combination;then,we give the related concepts ofq-torsional rigidity,corre-spondingL_p-Minkowski problem and basic facts,such as-torsional rigidity measure,L_p-torsional rigidity measure,theL_p-Minkowski problem of-torsional rigidity and its Minkowski inequality.In Chapter 3,we focus on the uniqueness of theL_p-Minkowski problem of-torsional rigidity.First of all,we obtain the Brunn-Minkowski inequality in the sense ofL_plinear combination of-torsional rigidity,and the conditions of the equation are obtained;then,we get theL_p-Hadamard variational formula(first-order variational formula)of-torsional rigidity for convex body in smooth case,and derive theL_p-Minkowski inequality of-torsional rigidity;finally,we obtain the uniqueness of theL_p-Minkowski problem ofq-torsional rigidity by using theL_p-Minkowski inequality ofq-torsional rigidity.The last part is the summary of the full text,which summarizes the research results and innovation of our works.