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Finding the Minimum of a Functional

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For which curve u(x) does the functional L[u], defined as the integral of

F[x, u, u'] = (1/2) (u')^2 + u u' + u + u',

attain a minimum when the values of u(x) are not specified at the end points of the interval of definition [a, b]?

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Solution Summary

The problem of finding the minimum of a specific functional with free boundary conditions is considered. A family of solutions is first obtained by calculating the solution to the corresponding Euler-Lagrange equation. Conditions under which the resulting two-parameter family of extremal solutions is minimal are then considered.

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