Existence of Solutions to Some Classical Variational Problems
Some non-coercive variational integrals are considered, including the classical time-of-transit functional arising in the problem of the brachistochrone, and the area functional in the problem of the minimal surface of revolution. A minimizer is construct
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Abstract Some non-coercive variational integrals are considered, including the classical time-of-transit functional arising in the problem of the brachistochrone, and the area functional in the problem of the minimal surface of revolution. A minimizer is constructed by means of the direct method. More precisely, each admissible curve is replaced by its convex envelope, and the functional is shown to decrease. Hence, there exists a minimizing sequence made up of convex curves, which in turn possesses a locally uniformly converging subsequence. The limiting curve is a minimizer because the functionals under consideration are continuous under such a convergence. Keywords Brachistochrone · Convex envelope · Direct method
1 Introduction Galileo compared the time of descent along an arc of circle with that of a corresponding chord, and also with a piecewise-smooth curve made up of two chords, concluding that brevissimo sopra tutti i tempi sarà quello della caduta per l’arco (shortest among all times will be that of the fall along the arc) [14, Fourth Day]. However, there exist curves whose time of transit is smaller. The classical problem of the brachistochrone consists in finding a curve γ0 joining two given points A, B in a vertical plane, so that a point mass placed at rest in A, constrained to γ0 and subject to a uniform gravity field reaches B in the shortest time. The effect of friction is neglected. More generally, the point mass may be given an initial velocity whose modulus v0 ≥ 0 is prescribed. The solution γ0 is called brachistochrone from the Greek βραχιςτ oσ (shortest) and χρoνoσ (time) [3]. Together with Newton’s problem of minimal resistance ([17, 18], [20, pp. 1–3]), the problem of the brachistochrone is considered as one of the starting problems of the calculus of variations. It was posed by Johann Bernoulli A. Greco (B) Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, via Ospedale 72, 09124 Cagliari, Italy e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_9, © Springer-Verlag Italia 2013
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in 1696 as a mathematical defy, and then solved by himself and some of his contemporaries. From a modern point of view, they characterized any possible solution, but underestimated the importance of an existence proof. Details are found in several textbooks [1, 3, 11, 13, 22]. Proving existence of the brachistochrone is more delicate: the result is often achieved by means of the concept of field of extremals and related theory, which has its roots in the work of Weierstrass (cf. [3, 12, 21]). Alternative approaches are found in [4, 15] and in the present paper. Let us recall a well-known mathematical model of the problem. Let I = (a, b) be a nonempty, bounded, open interval on the real line, and denote by X the class of all functions u in the Sobolev space W 1,1 (I ) attaining prescribed boundary values u(a) = 0 and u(b) = ub ≤ 0. Here and in the sequel, each
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