Invariance of solutions to invariant nonparametric variational problems

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 $f \circ g|B = f|B$ 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.