Research On The Functional Of Progressive Lens Design And Helfrich Functional

In this thesis,a broader class of functionals including Willmore functionals,Gauss curvature and area functionals are investigated,that are progressive lens design functionals and Helfrich functionals.The former is the model proposed when designing reading glasses lenses,which is currently the most widely used model in the market.And the latter is proposed when studying the problem of minimal energy in biological and chemical substances.This thesis mainly discuss the minimiser(function that reaches the minimum)of progressive lens design functionals and Helfrich functionals under Dirichlet boundary value conditions.The main research results in this paper include the following parts:1.We prove the existence of minimiser of progressive lens design functionals in the class of desirable functions under surfaces of revolution.The Euler-Lagrange equation corresponding to its functional variational problem is a fourth-order nonlinear elliptic partial differential equation.After linearizing the equation,we obtain the existence and uniqueness of the solution of the Dirichlet boundary value problem for the equation.Furthermore,we demonstrate the analyticity of the solution by using a easier way.2.We show the existence and regularity of the minimiser to a special Helfrich functional under the surface of revolution,in the more general class of desirable functions.And the result of the symmetry of the minimiser is also obtained.Meanwhile,we prove some properties of the minimiser,such as monotonicity,asymptotics,etc.,and partially answer an open question in the literature[49].The research results showes,to the general Helfrich functional,the existence and regularity of the minimiser in the general class of desirable functions under the surface of revolution are also valid.3.The minimiser of the Willmore functional under the Dirichlet boundary value condition does not necessarily exist within its classical definition range to a general twodimensional surface,so it was extended in literature[50].For progressive lens design functional and Helfrich functional,which is more general compared with the Willmore functional,we also discuss the existence of minimiser of relaxed progressive lens design functional and Helfrich functional after its extension under Dirichlet boundary value condition.