Isoperimetric problems as well as free minimum problems for non-coercive, one-dimensional functionals are investigated. Models are the classical problem of the catenary, the brachistochrone, and Newton's problem of minimal resistance. A convex minimizer is shown to exist in classes of competing functions containing both convex and non-convex elements. All problems are solved by means of convex rearrangement, a transformation that turns an absolutely continuous function into a convex one having same boundary values and same graph length but less energy.
Minimization of non-coercive integrals by means of convex rearrangement
GRECO, ANTONIO
2012-01-01
Abstract
Isoperimetric problems as well as free minimum problems for non-coercive, one-dimensional functionals are investigated. Models are the classical problem of the catenary, the brachistochrone, and Newton's problem of minimal resistance. A convex minimizer is shown to exist in classes of competing functions containing both convex and non-convex elements. All problems are solved by means of convex rearrangement, a transformation that turns an absolutely continuous function into a convex one having same boundary values and same graph length but less energy.
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