Comparison Principles for General Potential Theories and PDEs

One main purpose of this research monograph is to prove comparison principles for nonlinear potential theories in euclidean spaces in a very straightforward manner from duality and monotonicity. We shall also show how to deduce comparison principles for nonlinear differential operators -- a program seemingly different from the first. However, we shall marry these two points of view, for a wide rivaety of equations, under something called thecorrespondence principle. In potential theory one is given a constraint set on the 2-jets of a function, and the boundary of the constraint set gives a differential equation. There are many differential operators, suitably organized around the constraint set, which give the same equation. So potential theory gives a great strengthening and simplification to the operator theory. Conversely, the set of operators associated to the constraint set can have much to say about the potential theory. An object of central interest here is that of monotonicity, which explains and unifies much of the theory. We shall always assume that the maximal monotonicity cone for a potential theory has interior. This is automatic for gradient-free equations where monotonicity is simply the standard degenerate ellipticity (positivity) and properness (negativity) assumptions. We show that for each such potential theory there is an associated canonical operator F, defined on the entire 2-jet space and having all the desired properties. Furthermore, comparison holds for this F on any domain which admits a classical strictly M-subharmonic function, where M is a monotonicity subequation for the constrain set defining the potential theory. For example, for the potential theory corresponding to convex functions, the canonical operator is the minimal eigenvalue of the Hessian. On the operator side there is an important dichotomy into the unconstrained cases and constrained cases, where the operator must be restricted to a proper subset of 2-jet space. The unconstrained case is best illustrated by the canonical operators, whereas the constrained case is best illustrated by Dirichlet-Garding operators. The amonograph gives many, many examples from pure and applied mathematics, and also from theoretical physics.