Convexity criteria and uniqueness of absolutely minimizing functions

TL;DR

This paper establishes uniqueness of absolutely minimizing functions for convex Hamiltonians based on boundary data, extending key equivalences in the calculus of variations in L-infinity.

Contribution

It generalizes the conditions under which absolutely minimizing functions are uniquely determined, broadening the theoretical framework in the calculus of variations in L-infinity.

Findings

Uniqueness of absolutely minimizing functions under minimal assumptions.

Extension of equivalences between comparison with cones, convexity, and minimizing properties.

Advancement in the theoretical understanding of the calculus of variations in L-infinity.

Abstract

We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary values under minimal assumptions on $H.$ Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in $L^\infty.$