The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry

Calculus of variations is a fundamental mathematical discipline focused on optimizing functionals, which map sets of functions to real numbers. This field is essential for numerous applications, including the formulation and solution of partial differential equations (PDEs) and the study of differential geometry. In PDEs, calculus of variations provides methods to find functions that minimize energy functionals, leading to solutions of various physical problems. In differential geometry, it helps understand the properties of curves and surfaces, such as geodesics, by minimizing arc-length functionals. This paper explores the intrinsic connections between these areas, highlighting key principles such as the Euler-Lagrange equation, Ekeland’s variational principle, and the Mountain Pass theorem, and their applications in solving PDEs and understanding geometrical structures.