On The Second Boundary Value Problem For A Class Of Special Lagrangian Curvature Potential Equation

In calibration theory,the existence of minimal Lagrangian submanifolds has been a topic of interest,and the existence of minimal Lagrangian diffeomorphism between two smooth bounded regions can be translated into the second boundary value problem for special Lagrangian equation.Therefore,it is of great theoretical and practical importance to study its mathematical theory.In this thesis,we consider the second boundary value problem for special Lagrangian curvature potential equation.First,we obtain the corresponding the strict obliqueness estimate and the second derivative estimate of the solution of the equation by constructing auxiliary functions based on the properties of the equations.In particular,this thesis uses Urbas’ technique which considers the covariant derivatives of the second fundamental form of the graph of the solution of the equation in a local orthonomal frame field,and thus exploits the relationship between the second fundamental form of the graph’ the normal curvature and the second order derivatives of the solution of the equation to obtain the second derivative estimate of the solution.Finally,considering the family of problems with bounded intervals,we prove that the family of problems satisfies the structural conditions of the original operators,so as to obtain the strict obliqueness estimates and the second derivative estimates of the solutions corresponding to the family of problems,and then obtain the existence of the solution by the standard continuity method.