A uniqueness theorem for \(Z^ n\)-periodic variational problems
Let \(F: R^ n\times R\times R^ n\to R\) be an appropriate integrand which is Z-periodic in the first \(n+1\) variables. Moser's variational problem consists in finding functions \(u: R^ n\to R\) which minimize \(\int F(x,u,u_ x)dx\) with respect to all compactly supported variations of u. Minimal solutions without self-intersections of this problem have a uniquely determined ``slope'' \(\alpha \in R^ n\). For irrational \(\alpha\) these minimal solutions are obtained as limits of minimal solutions with rational slope. The author describes a new approach to the basic invariant, the slope of a minimal solution without selfintersections, and shows that for every \(\alpha\in R^ n\setminus Q\) n the limit set of minimal solutions is essentially unique.
Minimal solutions without self- intersections
scientific article; zbMATH DE number 4032482