Invariance of solutions to invariant nonparametric variational problems

Abstract

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