Existence of Solutions to Some Classical Variational Problems

摘要: Some non-coercive variational integrals are considered, including the classical time-of-transit functional arising in problem of brachistochrone, and area minimal surface revolution. A minimizer is constructed by means direct method. More precisely, each admissible curve replaced its convex envelope, shown to decrease. Hence, there exists a minimizing sequence made up curves, which turn possesses locally uniformly converging subsequence. The limiting because functionals under consideration continuous such convergence.