Harmonic Vector Fields And Partial Differential Systems Involving Operator Curl

This dissertation is devoted to studying the harmonic vector fields4 and two partial differential systems involving curl operator,including Maxwell equations and Lam(?) equations.We mainly study the asymptotic behavior of the solution of the equations involving curl operator.For Maxwell equations,we study the asymptotic behavior of solution in varying domains.Harmonic vector fields are closely related to the solution of Maxwell equations,so we first study of the decomposition and asymptotic behavior of harmonic vector fields.For Lam(?) equations,we study the asymptotic behavior of the solution as the parameter changes.In the first chapter,we briefly introduce our research motivations and state the main results.In Chapter 2,we study the harmonic vector fields on the domain with holes5.We decompose the harmonic vector fields,and obtain the asymptotic behavior of the harmonic fields as the number of holes tends to infinity.In Chapter 3,we study the asymptotic behavior of solutions of Maxwell equations in varying domains.Mainly,we show the following situations:(1)We consider the Maxwell equations in a domain with holes.We prove that when the radius of the holes tends to zero and the number of holes tends to infinity,the solution can be decomposed into two parts,where one is the Dirichlet vector field and the other part converges to the solution of the Maxwell equations in the domain without holes6;(2)We consider the Maxwell equations in a cylinder.We prove that when the length of the cylinder goes to infinity,the solution converges to a solution of the Maxwell equations in the infinite cylinder with the rate of convergence being exponential.In Chapter 4,we study Lam(?) equations with a small parameter.Asymptotic behaviors of two Lam(?) equations with different boundary conditions are examined.We prove that the solutions converge to the solutions of two degenerate elliptic equations involving operator curl with different boundary conditions respectively as the parameter tends to zero.