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We prove comparison principles for nonlinear potential theories in euclidianspaces in a very straightforward manner from duality and monotonicity. We shallalso show how to deduce comparison principles for nonlinear differentialoperators, a program seemingly different from the first. However, we shallmarry these two points of view, for a wide variety of equations, undersomething called the correspondence principle. In potential theory one is givena constraint set F on the 2-jets of a function, and the boundary of F gives adifferential equation. There are many differential operators, suitablyorganized around F, which give the same equation. So potential theory gives agreat strengthening and simplification to the operator theory. Conversely, theset of operators associated to F can have much to say about the potentialtheory. An object of central interest here is that of monotonicity, whichexplains and unifies much of the theory. We shall always assume that themaximal monotonicity cone for a potential theory has interior. This isautomatic for gradient-free equations where monotonicity is simply the standarddegenerate ellipticity and properness assumptions. We show that for each suchpotential theory F there is an associated canonical operator, defined on theentire 2-jet space and having all the desired properties. Furthermore,comparison holds for this operator on any domain which admits a regularstrictly M-subharmonic function, where M is a monotonicity subequation for F.On the operator side there is an important dichotomy into the unconstrainedcases and constrained cases, where the operator must be restricted to a propersubset of 2-jet space. These two cases are best illustrated by the canonicaloperators and Dirichlet-Garding operators, respectively. The article givesmany, many examples from pure and applied mathematics, and also fromtheoretical physics.

Comparison principles by monotonicity and duality for constant coefficient nonlinear potential theory and PDEs