Some monotonicity results for minimizers in the calculus of variations

TL;DR

This paper establishes monotonicity properties for minimizers and stable solutions of general energy functionals in the calculus of variations, leading to rigidity results in low dimensions under quadratic growth conditions.

Contribution

It introduces new monotonicity results for minimizers of energy functionals with quadratic growth assumptions, extending understanding of solution structure.

Findings

Monotonicity properties for minima and stable solutions

Rigidity results for global solutions in low dimensions

Conditions under which solutions exhibit monotonicity

Abstract

We obtain monotonicity properties for minima and stable solutions of general energy functionals of the type $$ \int F(\nabla u, u, x) dx $$ under the assumption that a certain integral grows at most quadratically at infinity. As a consequence we obtain several rigidity results of global solutions in low dimensions.