In this paper we study the existence of discontinuous, bounded variation optimal solutions of variational problems for simple integrals of the Lagrange type with fixed ends. The integrand is a continuous function which satisfies a growth condition. The Serrin integral for bounded variation functions is obtained by a lower limit process from the usual Lebesgue integral when a bounded variation curve is approximated by a sequence of absolutely continuous trajectories. Using a semicontinuity theorem for the cost functional, an existence theorem is formulated and some estimation relations between the Serrin and Lebesgue integrals are proved in the case of a single discontinuity point. Next these problems are discussed for length integrals with a positive, C1-weight φ(⋅), i.e. the integrand is now φ(t)[1+(x′(t))2]1/2. The last part is devoted to an integral functional involving the integrand |x′(t)−f(t,x(t))|, where f is a continuous, sublinear function. The existence of either absolutely continuous or discontinuous, bounded variation solutions is discussed in detail.

Remarks on discontinuous optimal solutions for simple integrals of the Calculus of Variations

PUCCI, Patrizia
1989

Abstract

In this paper we study the existence of discontinuous, bounded variation optimal solutions of variational problems for simple integrals of the Lagrange type with fixed ends. The integrand is a continuous function which satisfies a growth condition. The Serrin integral for bounded variation functions is obtained by a lower limit process from the usual Lebesgue integral when a bounded variation curve is approximated by a sequence of absolutely continuous trajectories. Using a semicontinuity theorem for the cost functional, an existence theorem is formulated and some estimation relations between the Serrin and Lebesgue integrals are proved in the case of a single discontinuity point. Next these problems are discussed for length integrals with a positive, C1-weight φ(⋅), i.e. the integrand is now φ(t)[1+(x′(t))2]1/2. The last part is devoted to an integral functional involving the integrand |x′(t)−f(t,x(t))|, where f is a continuous, sublinear function. The existence of either absolutely continuous or discontinuous, bounded variation solutions is discussed in detail.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/117431
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