Invariance of solutions to invariant nonparametric variational problems

Author:
John E. Brothers

Journal:
Trans. Amer. Math. Soc. **264** (1981), 91-111

MSC:
Primary 49F22; Secondary 35J20

DOI:
https://doi.org/10.1090/S0002-9947-1981-0597869-X

MathSciNet review:
597869

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Abstract: Let be a weak solution to the Euler-Lagrange equation of a convex nonparametric variational integral in a bounded open subset of . Assume the boundary of to be rectifiable. Let be a compact connected Lie group of diffeomorphisms of a neighborhood of which leave invariant and assume the variational integral to be -invariant. Conditions are formulated which imply that if is continuous on and for then for every . If the integrand is strictly convex then can be shown to have a local uniqueness property which implies invariance. In case is not strictly convex the graph of in is interpreted as the solution to an invariant parametric variational problem, and invariance of , hence of , follows from previous results of the author. For this purpose a characterization is obtained of those nonparametric integrands on which correspond to a convex positive even parametric integrand on in the same way that the nonparametric area integrand corresponds to the parametric area integrand.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0597869-X

Keywords:
Nonparametric integrand,
parametric integrand,
invariant integrand,
Euler-Lagrange equation,
weak solution

Article copyright:
© Copyright 1981
American Mathematical Society