Analogue of Pontryagin's maximum principle for multiple integrals minimization problems

TL;DR

This paper extends Pontryagin's maximum principle to multiple integral minimization problems, showing the maximum is achieved over rank-one matrices rather than all matrices, with illustrative examples.

Contribution

It introduces a novel version of Pontryagin's maximum principle where the maximization is restricted to rank-one matrices for multiple integrals.

Findings

Maximum achieved over rank-one matrices

Theoretical proof of the extended principle

Examples illustrating the new maximum condition

Abstract

The theorem like Pontryagin's maximum principle for multiple integrals is proved. Unlike the usual maximum principle, the maximum should be taken not over all matrices, but only on matrices of rank one. Examples are given.