In this paper we study the existence of a solution in Lloc() to the Euler–Lagrange equation for the variational problem infu+W01()(ID(u)+g(u))dx(01) with D convex closed subset of Rn with non empty interior. By means of a disintegration theorem, we next show that the Euler–Lagrange equation can be reduced to an ODE along characteristics, and we deduce that there exists a solution to Euler–Lagrange different from 0 a.e. and satisfies a uniqueness property. These results prove a conjecture on the existence of variations on vector fields stated in Bertone and Cellina (On the existence of variations).
On the Euler-Lagrange equation for a variational problem: the general case II / Bianchini, S.; Gloyer, Matteo. - In: MATHEMATISCHE ZEITSCHRIFT. - ISSN 0025-5874. - 265:4(2010), pp. 889-923. [10.1007/s00209-009-0547-2]
On the Euler-Lagrange equation for a variational problem: the general case II
Bianchini, S.;Gloyer, Matteo
2010-01-01
Abstract
In this paper we study the existence of a solution in Lloc() to the Euler–Lagrange equation for the variational problem infu+W01()(ID(u)+g(u))dx(01) with D convex closed subset of Rn with non empty interior. By means of a disintegration theorem, we next show that the Euler–Lagrange equation can be reduced to an ODE along characteristics, and we deduce that there exists a solution to Euler–Lagrange different from 0 a.e. and satisfies a uniqueness property. These results prove a conjecture on the existence of variations on vector fields stated in Bertone and Cellina (On the existence of variations).| File | Dimensione | Formato | |
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