Some Topics On L_p Minkowski Problems

In this thesis,we focus on two Lp Minkowski problems associated to q-Capacity and torsional rigidityIn Section 1,we list the background on this topic,main results of the present paper and some lemmas which are needed in the present paperIn Section 2,we introduce the concepts of Lp mixed q-Capacity,Lp q-Capacity measure and propose a Lp Minkowski problem for q-Capacity for any fixed n?2,q? n and p>1.We prove a Lp Minkowski inequality for q-Capacity and an isopermetric inequality for q-Capacity and then prove an Hadamard variational formula for q-Capacity under p-sum via Lp Minkowski inequality for q-Capacity.As an application,we prove the existence of solutions to the Lp Minkowski problem mentioned above via the variational method.The uniqueness of solutions to the Lp Minkowski problem mentioned above is also proved by the Lp Minkowksi inequalityIn Section 3,we introduce the concepts of Lp mixed torsional rigidity,Lp torsion measure and propose a Lp Minkowski problem for torsional rigidity for any fixed p>1.We prove a Lp Minkowski inequality for torsional rigidity and an isopermetric inequality for torsional rigidity and then prove an Hadamard variational formula for torsional rigidity under p-sum via Lp Minkowski inequal-ity for torsional rigidity.As an application,we prove the existence of solutions to the Lp Minkowski problem mentioned above via the variational method.The uniqueness of solutions to the Lp Minkowski problem mentioned above is also proved by the Lp Minkowksi inequalityIt is worth mentioning that the Lp Minkowski inequality and isopermetric inequality for q-Capacity and torsional rigidity have been proved and a new proof of the Hadamard variational formula for q-Capacity and torsional rigidity have been given.