[Paper Review] Sublinear Elliptic Operators

Karl K. Brustad|arXiv (Cornell University)|Feb 12, 2018

Numerical methods in inverse problems1 references3 citations

TL;DR

This paper establishes a comprehensive theory for second-order sublinear elliptic operators by linking them to convex bodies in the space of symmetric matrices. It proves that every sublinear operator corresponds uniquely to a convex body, enabling a geometric characterization of key properties like degenerate and uniform ellipticity, with applications to Pucci and dominative p-Laplacian operators through extremal point representations.

ABSTRACT

We investigate second order elliptic equations \[F(\mathcal{H}u) = 0\] where the function $F\colon S(n) o\mathbb{R}$ on the space of symmetric $n imes n$ matrices is assumed to be sublinear. There is very little to be found in the literature devoted particularly to sublinear elliptic operators. When examples of such operators occur, they are often merely treated as members of the larger class of convex operators. That class has been thoroughly investigated and many of its aspects are well understood. There is, however, something to be said about sublinear operators that do not, in general, apply to convex operators.

Motivation & Objective

• To develop a systematic theory for sublinear elliptic operators, which are underexplored despite their importance in nonlinear PDEs.
• To establish a one-to-one correspondence between sublinear operators and convex bodies in the space of symmetric matrices.
• To characterize degenerate and uniformly elliptic sublinear operators using geometric properties of their associated convex bodies.
• To derive comparison principles, superposition rules, and fundamental solutions for sublinear operators.
• To extend known results on viscosity solutions and regularity to the sublinear setting, particularly for the Pucci and dominative p-Laplacian operators.

Proposed method

• Represent sublinear operators as F(X) = max_{Y∈K} ⟨Y,X⟩, where K is a convex body in S(n), linking operator theory to convex geometry.
• Use the extreme point structure of convex bodies to characterize the Pucci operator as P_{λ,Λ}(X) = max_{P∈Pr(n)} tr((λI + (Λ−λ)P)X), yielding a closed-form expression.
• Prove that uniform ellipticity is equivalent to the inclusion K ⊆ K_{λ,Λ}, where K_{λ,Λ} is the convex body associated with the Pucci operator.
• Apply the theory to show that viscosity solutions of uniformly elliptic sublinear equations are C^{2,α} regular.
• Use the duality between F(X) = max_{Y∈K} ⟨Y,X⟩ and the support function of K to derive comparison principles and stability results.
• Utilize sup/inf-convolutions and viscosity solution techniques to analyze subsolutions and supersolutions, noting their asymmetric behavior under sign reversal.

Experimental results

Research questions

• RQ1 How can sublinear elliptic operators be systematically classified and characterized using convex geometry in the space of symmetric matrices?
• RQ2 What is the precise geometric condition on the convex body K ⊂ S(n) that ensures degenerate or uniform ellipticity of the associated operator F(X) = max_{Y∈K} ⟨Y,X⟩?
• RQ3 Why do subsolutions and supersolutions behave asymmetrically under sign reversal in the sublinear setting, unlike in linear or p-Laplacian cases?
• RQ4 How can the Pucci operator and the dominative p-Laplacian be represented as maxima over extremal matrices in a convex body?
• RQ5 What is the role of the extreme point structure of the convex body in determining the regularity and comparison properties of solutions?

Key findings

• Every sublinear operator F: S(n) → ℝ arises uniquely from a convex body K ⊂ S(n), with F(X) = max_{Y∈K} ⟨Y,X⟩.
• The Pucci operator P_{λ,Λ}(X) is characterized as P_{λ,Λ}(X) = max_{P∈Pr(n)} tr((λI + (Λ−λ)P)X), which simplifies to Λ tr(X⁺) − λ tr(X⁻).
• A sublinear operator is uniformly elliptic with constants 0 < λ ≤ Λ if and only if its associated convex body K is contained in K_{λ,Λ}, the body corresponding to the Pucci operator.
• Viscosity solutions to uniformly elliptic sublinear equations are C^{2,α} regular, extending classical results from convex operators to the sublinear case.
• The set of extreme points of the convex body K_{λ,Λ} is exactly {λI} ∪ (Λ−λ)Pr(n), which corresponds to the orthogonal projections and identity matrix.
• Subsolutions and supersolutions are not symmetric under sign reversal: if u is a supersolution, then −u is a subsolution, but the converse does not hold in general.