Application Of Calculus Of Variations In The Plateau Problem And Optimal Transportation Problem

Calculus of variations are the field of mathematics that deals with functionals,as opposed to ordinary calculus that deals with functions.In this field,one usually looks for the extremal functions:they are maximizer or minimizer of a given functional.It has important applications in other branches of pure mathema tics,computational mathematics,probability theory,optimal control,quantum physics,material science and other scientific fields.In this master thesis,we mainly concern the application of calculus of variations in geometric analysis and probability.In particular,we will apply the direct method in the calculus of variations to the well-known Plateau problem,and Kantorovich duality to solve the optimal transport problem.The famous Plateau problem can be briefly described as follows:Given a closed curveΓ in the Euclidean space Rn,whet her there is a minimal surface bounded by Γ.This problem was first formulated in 1760 by the famous mathematician Lagrange.Subsequently,the Belgian physicist Plateau conducted a large number of soap bubble experiments in the 19th century,resulting in the Plateau law associated with minimal surfaces.Around 1930,Douglas and Rado independently proved the presence of minimal surfaces bounded by Jordan curves.Douglas won the Fields medal in 1936 for this work.In 1948,Morrey further generalized the results of Douglas and Rado to homogeneous Riemannian manifolds.In a recent work,Lytchak-Wenger solved the Plateau problem within the framework of proper metric spaces,and successfully applied the results to geometric group theory,metric geometry,and non-smooth analysis.The optimal transportation problem is to study how to minimize the transportation cost when a given cargo is transported from one place to another.This problem was first proposed by the French mathematician Monge in 1781.Then the former Soviet mathematician and economist.Kantorovich solved the Mongc optimal transport problem.Because of its importance in economics,Kantorovich won the Nobel Prize in Economics in 1975 Award.Very recently,Figalli won the Fields medal in 2018 for his research on the mathematical theory related to optimal transportation.In addition to pure mathematics and economics,the theory of optimal transportation has also been widely used in many engineering fields.In this master thesis,we will give detailed proofs of Douglas-Rado in Euclidean space,Lytchak-Wenger in proper metric space,and Kantorovich’s solution of Monge’s optimal transportation problem.Therefore,the thesis is naturally divided into three parts.In the first part,we consider that given a simple closed curve Γ in a Euclidean space.how to find a minimal surface bounded by Γ.Due to the lack of compactness caused by the intrinsic invariance of the area functional,we cannot directly apply the direct method in calculus of variations.Following Douglas’ original idea,we turned to minimizers of Dirichlet’s energy,applying the compactness theory of harmonic function to overcome the obstacle,thus solving the Plateau problem.In the second part,we will solve the Plateau problem in the framework of proper metric spaces.Unlike Euclidean spaces,there is no compactness theory of harmonic functions that can be applied directly due to the lack of smooth differentiable structures.We follow the idea of Lytchak-Wenger,and apply the theory of metric space valued Sobolev maps to overcome this difficulty.We will also demonstrate the local Holder regularity and global regularity of solutions in those metric spaces that support a quadratic isoperimetric inequality.In the third part,we first study the Kantorovich problem,and use the direct method in the calculus of variations to prove the existence of minimizers.Secondly,we discuss the Kantorovich duality property.The duality formula is extended to the lower semi-continuous cost function.Finally,we prove the necessary and sufficient conditions for the optimality of the lower semi-continuous cost function.