Abstract
This chapter focuses on the convexity and calculus of variations. A sufficiently smooth minimizer satisfies the Euler–Lagrange differential equation. A real-valued function is convex if its epigraph is convex. The lower semicontinuity of the functionals that are considered in the calculus of variations is implied by a convexity condition. The direct method of the calculus of variations has great significance in the numerical treatment of variational problems and partial differential equations. The chapter discusses multiple integrals in the calculus of variations, and it describes lower semicontinuity and existence.
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