Regularity of solutions to higher-order integrals of the calculus of variations

TL;DR

This paper establishes new regularity conditions for higher-order calculus of variations problems, ensuring minimizers have bounded derivatives and preventing the Lavrentiev phenomenon.

Contribution

It introduces novel regularity criteria for higher-order variational problems, guaranteeing boundedness of minimizers' derivatives and eliminating the Lavrentiev gap.

Findings

Minimizers have essentially bounded derivatives under new conditions

Lavrentiev phenomenon does not occur with these regularity criteria

Results apply to autonomous integral functionals with coercive derivatives

Abstract

We obtain new regularity conditions for problems of calculus of variations with higher-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main regularity result asserts that autonomous integral functionals with a Lagrangian having coercive partial derivatives with respect to the higher-order derivatives admit only minimizers with essentially bounded derivatives.