Regularity of Solutions to Second-Order Integral Functionals in Variational Calculus

TL;DR

This paper establishes new regularity conditions for second-order variational problems, ensuring minimizers have bounded derivatives and preventing the Lavrentiev phenomenon in certain autonomous integral functionals.

Contribution

It introduces novel regularity criteria for second-order calculus of variations problems, demonstrating boundedness of minimizers and eliminating the Lavrentiev phenomenon under specified conditions.

Findings

Minimizers have essentially bounded derivatives under new regularity conditions.

Lavrentiev phenomenon does not occur for the studied class of problems.

Superlinearity of the Lagrangian's derivatives is key to regularity results.

Abstract

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