The Study Of Timelike Extremal Surface In Minkowski Space

The present Ph.D dissertation is devoted to the study of time-like extremal surface in Minkowski space.We prove the global existence of classical solutions of Cauchy problem and the mixed initial-boundary value problem for equation of time-like extremal surface and gets the asymptotic behavior of global classical solutions.In the study of physics and other application areas,mosdy the problem can be reduced to the mathematical problem.As an important and non-trivial model in string theory and particle physics,extremal surface in Minkowski space is also an important investigation topic in mathematics.It also plays an important role in fluids mechanics, electromagnetism as well as in the theory of black hole.Extremal surface equation is Euler-Lagrange equation of its area functional,which is a first order system of conservation laws.Using the characteristic methods of quasilinear hyperbolic systems,we consider the related problem of the equation of time-like extremal surface in Minkowski space R^1+(1+n),i.e.Cauchy problem,the mixed initial-boundary value problem(including Dirichelt boundary problem,Neumann boundary problem and Robin boundary problem) and get global classical solutions and asymptotic behavior with more interesting phenomena. It also plays a unique role in our understanding in geometry and physics by studying high dimensional time-like extremal surface in Minkowski space.The dissertation is organized as follows.Chapter One is an introduction.It is devoted to introducing physical background and previous mathematical research works on extremal surface,especially important achievements by other scholars who treated similar problems in quasilinear hyperbolic systems or global classical solutions and its asymptotic behavior related to those of ours that are presented in the following chapters.The main problems we concerned,main results we obtained,and methods we utilized in this Ph.D.dissertation are also illustrated with com- ments.In Chapter Two we concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields.Based on the existence results of global classical solutions,we prove that when t tends to infinity,the solution approaches a combination of C¹ travelling wave solutions,provided that L¹âˆ©L^∞norm of the initial data as well as its derivative are bounded.Application is given to equation of the time-like extremal surface in Minkowski space.In Chapter Three we investigate the mixed initial-boundary value problem for the equation of time-like extremal surface in Minkowski space R^1+(1+n) in the first quadrant. Under the assumptions that the initial data are bounded and the boundary data are small, we prove the global existence and uniqueness of the C² solutions of the Dirichlet problem and Neumann problem for this kind of equation.Based on the existence results on global classical solutions,we show that:as t tends to infinity,the first order derivatives of the solutions approach C¹ travelling wave solution,provided that L¹âˆ©L^∞norm of the first and second order derivatives of the initial and boundary data are bounded.In which we can proved that geometrically this extremal surface is a generalized cylinder.Moreover, The travelling wave solutions are the exact solutions of system under consideration.This theory can also be used to study the equation of Born-Infeld type.e.g.the equation of relativistic strings moving.In Chapter Four we study the mixed initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space R^1+(1+n) on the strip R⁺×[0,1].Under the assumptions that the boundary data are small and decaying,we get the global existence and uniqueness of classical solutions.In Chapter Five we show that the nonlinear wave equation corresponding to the high dimensional time-like extremal surface equation in Minkowski space have global smooth solutions for initial data sufficiently close to the arbitrary time-like plane a₀t+a₁x₁+...+ a_nx_n+b=0.