[Paper Review] Characterizing the Strong Maximum Principle

F. Reese Harvey, H. Blaine Lawson|arXiv (Cornell University)|Sep 6, 2013

Nonlinear Partial Differential Equations17 references3 citations

TL;DR

This paper provides a complete characterization of when the strong maximum principle (SMP) holds for constant coefficient, degenerate elliptic subequations F(D²u) = 0. It introduces a characteristic function f(y) derived from F and proves that the SMP holds if and only if the integral ∫₀⁺ dy/f(y) diverges, offering a sharp, computable criterion for SMP validity in the borderline case.

ABSTRACT

In this paper we characterize the degenerate elliptic equations F(D^2u)=0 whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum principle. We introduce an easily computed function f(t) for t > 0, determined by F, and we show that the strong maximum principle holds depending on whether the integral \int dy / f(y) near 0 is infinite or finite. This complements our previous work characterizing when the (ordinary) maximum principle holds. Along the way we characterize radial subsolutions.

Motivation & Objective

• To determine precisely when viscosity subsolutions of degenerate elliptic equations F(D²u) ≥ 0 satisfy the strong maximum principle (SMP).
• To extend prior geometric characterizations of the ordinary maximum principle (MP) to the stronger SMP in the context of constant coefficient, pure second-order operators.
• To identify a complete, computable criterion for SMP validity in the 'borderline' case where F(0) = 0 and the standard SMP fails or holds depending on the structure of F.
• To analyze radial subsolutions and their role in determining SMP behavior, particularly through a one-dimensional operator R↑f.
• To establish sufficient conditions for strong comparison principles and to examine exotic monotonicity subequations beyond cone structures.

Proposed method

• Define a characteristic function f(λ) = sup{μ : F(λPₑ⊥ − μPₑ) ≥ 0} for invariant F, which captures the radial behavior of subsolutions.
• Introduce a one-dimensional variable coefficient operator R↑fψ(t) = min{ψ′(t), ψ′′(t) + f(ψ′(t)/t)} that governs radial subsolutions.
• Prove that a radial function u(x) = ψ(|x|) is an F-subsolution if and only if ψ is an R↑f-subsolution, linking radial behavior to the characteristic function.
• Establish the key criterion: the SMP holds for borderline F if and only if ∫₀⁺ dy/f(y) = ∞, derived via analysis of increasing radial subsolutions.
• Use geometric and variational techniques, including the use of supporting functions and inf-convolution arguments, to analyze subsolutions near boundary points.
• Apply the theory to examples such as Pucci operators, Poincaré-type equations, and product subequations, demonstrating the criterion’s applicability.

Experimental results

Research questions

• RQ1 Under what precise conditions on a constant coefficient degenerate elliptic operator F does the strong maximum principle hold?
• RQ2 How does the behavior of radial subsolutions—specifically increasing radial functions—determine the validity of the SMP?
• RQ3 Can the SMP be characterized purely in terms of a single function f derived from F, independent of the full structure of F?
• RQ4 What is the role of invariance (e.g., O(n) or SU(n/2) symmetry) in simplifying the characterization of the SMP?
• RQ5 How does the integral condition ∫₀⁺ dy/f(y) = ∞ relate to the geometry of the subequation F and the existence of non-constant subsolutions with interior maxima?

Key findings

• The strong maximum principle holds for a constant coefficient, degenerate elliptic subequation F if and only if the integral ∫₀⁺ dy/f(y) diverges, where f is the characteristic function derived from F.
• For invariant F in the borderline case (f(0) = 0), the SMP holds precisely when the characteristic function f(y) decays slowly enough at y → 0⁺ for the integral to diverge.
• Radial subsolutions of F are completely characterized by the one-dimensional operator R↑f, which governs the behavior of ψ(|x|) for increasing ψ.
• The paper establishes a sufficient condition for strong comparison: if F satisfies the SMP, then subsolutions satisfy u ≤ v whenever u ≤ v on the boundary.
• The theory applies to important classes of equations, including Pucci operators and Poincaré-type equations, and reveals that some non-cone subequations can still satisfy the SMP.
• For cone subequations, the Riesz characteristic α_F = f(1) determines SMP validity: the SMP holds iff α_F < ∞, and α_Fα_F* ≥ 1 always holds.