We consider the problem of minimizing ∫<sub>a</sub><sup>b</sup> L(x(t),x′(t))dt, x(a) = A,x(b) = B. Under the assumption that the Lagrangian L is continuous and satisfies a growth assumption that does not imply superlinear growth, we provide a result on the relaxation of the functional and show that a solution to the minimum problem is Lipschitzian.
Cellina, A. (2004). The classical problem of the calculus of variations in the autonomous case: relaxation and Lipschitzianity of solutions. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 356(1), 415-426 [10.1090/S0002-9947-03-03347-6].
The classical problem of the calculus of variations in the autonomous case: relaxation and Lipschitzianity of solutions
CELLINA, ARRIGO
2004
Abstract
We consider the problem of minimizing ∫ab L(x(t),x′(t))dt, x(a) = A,x(b) = B. Under the assumption that the Lagrangian L is continuous and satisfies a growth assumption that does not imply superlinear growth, we provide a result on the relaxation of the functional and show that a solution to the minimum problem is Lipschitzian.
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