[Paper Review] Comparison principles by monotonicity and duality for constant coefficient nonlinear potential theory and PDEs

Marco Cirant, F. Reese Harvey+2 more|arXiv (Cornell University)|Sep 3, 2020

Geometric Analysis and Curvature Flows40 references4 citations

TL;DR

This paper establishes comparison principles for constant coefficient nonlinear potential theories and fully nonlinear PDEs using duality and monotonicity, introducing a correspondence principle that unifies potential theory and operator theory. The key contribution is a canonical operator $ F $ associated with each constraint set $ \mathcal{F} $, for which comparison holds on domains admitting a $ C^2 $ strictly $ \mathcal{M} $-subharmonic function, where $ \mathcal{M} $ is a monotonicity subequation for $ \mathcal{F} $. This framework simplifies and strengthens the analysis of degenerate elliptic equations and viscosity solutions.

ABSTRACT

We prove comparison principles for nonlinear potential theories in euclidian 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 variety of equations, under something called the correspondence principle. In potential theory one is given a constraint set F on the 2-jets of a function, and the boundary of F gives a differential equation. There are many differential operators, suitably organized around F, 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 F 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 and properness assumptions. We show that for each such potential theory F there is an associated canonical operator, defined on the entire 2-jet space and having all the desired properties. Furthermore, comparison holds for this operator on any domain which admits a regular strictly M-subharmonic function, where M is a monotonicity subequation for F. 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. These two cases are best illustrated by the canonical operators and Dirichlet-Garding operators, respectively. The article gives many, many examples from pure and applied mathematics, and also from theoretical physics.

Motivation & Objective

• To establish comparison principles for constant coefficient nonlinear potential theories using duality and monotonicity.
• To unify potential theory and operator theory through a correspondence principle that links constraint sets $ \mathcal{F} $ to differential operators.
• To define a canonical operator $ F $ on the full 2-jet space for each constraint set $ \mathcal{F} $, ensuring all desired properties are satisfied.
• To show that comparison holds for this canonical operator $ F $ on domains $ \Omega \subset \subset \mathbb{R}^n $ admitting a $ C^2 $ strictly $ \mathcal{M} $-subharmonic function.
• To demonstrate that monotonicity and duality provide a unifying and simplifying framework for degenerate elliptic equations and viscosity solutions.

Proposed method

• The paper uses duality theory for subequations to define $ \mathcal{F} $-subharmonic functions via constraint sets on 2-jets.
• It introduces the concept of a monotonicity cone $ \mathcal{M} $, with the maximal monotonicity cone assumed to have interior, ensuring structural stability.
• A canonical operator $ F $ is constructed on the full 2-jet space $ \mathbb{R} \times \mathbb{R}^n \times \mathcal{S}(n) $, derived from $ \mathcal{F} $, which captures the essential nonlinear PDE structure.
• Comparison principles are proven via the zero maximum principle for dual monotonicity cones and the use of upper test jets in viscosity theory.
• The proof relies on contradiction arguments using sequences of jets $ J_k \in \mathcal{F} $ converging to a limit $ J \notin \mathcal{F} $, exploiting the closedness of $ \mathcal{F} $.
• The correspondence principle links multiple operators to a single $ \mathcal{F} $, showing that potential theory simplifies and strengthens operator-theoretic analysis.

Experimental results

Research questions

• RQ1 How can comparison principles in nonlinear potential theory be derived directly from duality and monotonicity?
• RQ2 What is the role of the canonical operator $ F $ in unifying potential theory and operator theory for constant coefficient equations?
• RQ3 Under what geometric conditions on the domain $ \Omega \subset \subset \mathbb{R}^n $ does the comparison principle hold for the canonical operator $ F $?
• RQ4 How does the correspondence principle bridge the gap between constraint sets $ \mathcal{F} $ and the set of differential operators that yield the same PDE?
• RQ5 What is the significance of the maximal monotonicity cone having interior in ensuring the validity of comparison principles?

Key findings

• Comparison holds for the canonical operator $ F $ on any domain $ \Omega \subset \subset \mathbb{R}^n $ that admits a $ C^2 $ strictly $ \mathcal{M} $-subharmonic function, where $ \mathcal{M} $ is a monotonicity subequation for $ \mathcal{F} $.
• For the potential theory of convex functions, the canonical operator $ F $ is the minimal eigenvalue of the Hessian $ D^2u $ in the $ C^2 $-case.
• Existence of solutions to the Dirichlet problem is guaranteed in the constant coefficient case for $ C^2 $-smooth domains with strict boundary convexity, and uniqueness is equivalent to the comparison principle.
• The canonical operator $ F $ is well-defined on the entire 2-jet space and inherits all desired properties from $ \mathcal{F} $, ensuring consistency and robustness.
• The proof of the comparison principle relies on contradiction via sequences of jets $ J_k \in \mathcal{F} $ converging to $ J \notin \mathcal{F} $, which contradicts the closedness of $ \mathcal{F} $.
• The framework unifies unconstrained and constrained cases: canonical operators represent the unconstrained case, while Dirichlet-Gårding operators represent the constrained case.