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Transactions of the American Mathematical Society

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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
MathSciNet review: 597869
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Abstract: Let $ f$ be a weak solution to the Euler-Lagrange equation of a convex nonparametric variational integral in a bounded open subset $ D$ of $ {{\mathbf{R}}^n}$. Assume the boundary $ B$ of $ D$ to be rectifiable. Let $ D$ be a compact connected Lie group of diffeomorphisms of a neighborhood of $ D \cup B$ which leave $ D$ invariant and assume the variational integral to be $ G$-invariant. Conditions are formulated which imply that if $ f$ is continuous on $ D \cup B$ and $ f \circ g\vert B = f\vert B$ for $ g \in G$ then $ f \circ g = f$ for every $ g \in G$. If the integrand $ L$ is strictly convex then $ f$ can be shown to have a local uniqueness property which implies invariance. In case $ L$ is not strictly convex the graph $ {T_f}$ of $ f$ in $ {{\mathbf{R}}^n} \times {\mathbf{R}}$ is interpreted as the solution to an invariant parametric variational problem, and invariance of $ {T_f}$, hence of $ f$, follows from previous results of the author. For this purpose a characterization is obtained of those nonparametric integrands on $ {{\mathbf{R}}^n}$ which correspond to a convex positive even parametric integrand on $ {{\mathbf{R}}^n} \times {\mathbf{R}}$ in the same way that the nonparametric area integrand corresponds to the parametric area integrand.

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Keywords: Nonparametric integrand, parametric integrand, invariant integrand, Euler-Lagrange equation, weak solution
Article copyright: © Copyright 1981 American Mathematical Society

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