Subequation Subharmonics in Nonlinear PDEs

• Subequation subharmonics are a generalization of classical subharmonic functions that encode admissibility and degenerate ellipticity via a geometric, set-theoretic framework.
• They provide a rigorous basis for nonlinear potential theory, enabling singularity removal, oscillation estimates, and extensions to viscosity solution methods.
• This framework unifies potential theoretic approaches for nonlinear, elliptic PDEs, offering precise comparison principles, estimates, and stratification of singular sets.

Subequation subharmonics generalize the classical notion of subharmonic functions to encompass solutions and subsolutions of fully nonlinear elliptic partial differential equations (PDEs) by encoding admissibility and comparison principles into a geometric, set-theoretic framework. This theory enables rigorous analysis of nonlinear potential theory, removal of singularities, oscillation estimates, and provides a bridge to the viscosity theory for nonlinear PDEs.

1. Definitions: Subequations and Subequation Subharmonics

A subequation is a closed subset $F$ of the 2-jet bundle over a domain $X$ (written locally as $$X \times \mathbb{R} \times \mathbb{R}^n \times \text{Sym}^2(\mathbb{R}^n)$$) that encodes both the directionality and monotonicity constraints required of admissible subsolutions. Formally, $F$ must satisfy a positivity condition: $F + P \subset F,$ where $P$ is the set of jets with positive semidefinite Hessian (second derivative part), ensuring the “degenerate ellipticity” critical for nonlinear PDEs (Harvey et al., 2013, Harvey et al., 17 Sep 2025).

A function $u$ (usually upper semicontinuous) is defined to be $F$-subharmonic on $X$ if at every $x \in X$ and for every upper test function $\varphi$ tangent from above at $x$ (i.e., $u \leq \varphi$ near $x$, $u(x)=\varphi(x)$), the jet $J_x^2\varphi$ belongs to $F_x$: $J_x^{2,+}u \subset F_x.$ This condition means that $u$ is a “subsolution” to a fully nonlinear inequality constrained by $F$.

This setup unifies several familiar examples:

• Convex/Plurisubharmonic/Potential theory: For $F$ the cone of positive semidefinite Hessians, $F$-subharmonics are convex functions (Harvey et al., 2013, Harvey et al., 17 Sep 2025).
• Monge–Ampère, Gårding, and Hessian equations: The corresponding subequation $F$ encodes the positivity or monotonicity condition on the set of admissible Hessians as in the “Gårding cone” (Harvey et al., 13 Sep 2025).

2. The Correspondence Principle and Degenerate Ellipticity

The correspondence principle formalizes the equivalence between $F$-subharmonicity (in the potential-theoretic sense) and the standard notion of admissible viscosity subsolutions for degenerate elliptic (fully nonlinear) PDEs (Harvey et al., 17 Sep 2025):

• Given an operator $\mathcal{F}$, the associated subequation is $F = \{(x, J) \mid \mathcal{F}(x, J) \geq 0\}$; then $u$ is $F$-subharmonic if and only if $u$ is a viscosity subsolution of $\mathcal{F}(J_x^2u) = 0$, provided $\mathcal{F}$ is degenerate elliptic.

Degenerate ellipticity requires: $$\mathcal{F}(x, r, p, A) \leq \mathcal{F}(x, r, p, A+P), \qquad \forall P \geq 0$$ This is equivalent to the monotonicity of $F$ with respect to the cone $P$ in the 2-jet bundle, justifying the geometric encoding of admissibility.

For operators that are only elliptic on a restriction set $G$, admissibility is enforced by only allowing $F$-subharmonics to “live” on $G$. For example, for $\det D^2u = f(x)$ (the Monge–Ampère equation) with $f \geq 0$, admissibility is enforced by requiring $D^2u \geq 0$.

3. Removable Singularities and Polar Sets for Subequations

A cornerstone of the theory is the classification and removal of singularities for $F$-subharmonic functions (Harvey et al., 2013, Chu, 2016). The key concepts are:

• If $F$ is $M$-monotone, i.e. $F+M \subset F$ for some convex cone subequation $M$, and $E$ is a closed $M$-polar set (i.e., there exists a smooth $M$-subharmonic function $\psi$ such that $E = \{\psi = -\infty\}$), then any $F$-subharmonic function defined on $X \setminus E$ and locally bounded above near $E$ extends to a global $F$-subharmonic on $X$.
• The Riesz characteristic $p_M$ associated to $M$ quantifies the “size” of singular sets which can be removed, generalizing classical capacity and Hausdorff dimension results for harmonic and plurisubharmonic functions.

This provides removable singularity theorems for complex, quaternionic, $p$-plurisubharmonic, and more general geometrically defined subequations, extending classical theory to nonlinear and fully nonlinear contexts.

4. Quantitative and Geometric Stratification of Singular Sets

The structure of singularities of $F$-subharmonic functions is refined via quantitative stratification (Chu, 2016):

• For $F$-subharmonic $u$, the singular set $\mathcal{S}(u)$ can be stratified into layers according to the homogeneity of blow-up limits (tangent functions) at singular points. If $p$ is the Riesz characteristic of $F$, then $\dim_H (\mathcal{S}(u)) \leq n-p$.
• When homogeneity or uniqueness of tangents holds (e.g., geometrically determined subequations), sharper geometric measure-theoretic estimates (Minkowski content, rectifiability) are obtained for strata of singular points (based on the symmetry of tangents).

This stratification generalizes the classical understanding of regularity and singularities in PDE theory and geometric analysis, offering dimension estimates and structural insights far beyond the linear or harmonic case.

5. Potential-Theoretic Methods: Oscillation and Maximum Principles

Potential-theoretic approaches for fully nonlinear elliptic PDE via subequation subharmonics yield robust methods for a priori estimates and comparison principles (Harvey et al., 13 Sep 2025, Harvey et al., 17 Sep 2025):

• The Alexandrov–Bakelman–Pucci (ABP) estimate is extended to subequation subharmonics via the “determinant majorization” technique. For solutions (or supersolutions) $u$ to inhomogeneous equations $\mathfrak{g} (D^2u) = f(x)$ with admissibility constraint $D^2u \in \bar{\Gamma}$, the key estimate reads:

$$\sup_\Omega u \leq \sup_{\partial\Omega} u + \frac{\operatorname{diam}(\Omega)}{|B_1|^{1/n}\,\mathfrak{g}(I)^{1/N}} \left\|f^{1/N}\right\|_{L^n(\Omega)}.$$

where $\mathfrak{g}$ is a homogeneous polynomial operator (for example, determinant, $\sigma_k$, or more general Gårding–Dirichlet operators), and $N$ is its degree.

• Semiconvex approximation is pivotal: generic viscosity solutions are approximated by locally semiconvex subharmonic functions via sup-convolution, making classical differential inequalities and area formula arguments applicable.
• The duality-monotonicity-fiberegularity method leverages the algebraic and topological structure of the subequation and its dual to reduce the comparison of sub/superharmonic functions to the “zero maximum principle” in a monotonicity cone, allowing for direct proofs of uniqueness and stability.

These methods transport the powerful machinery of potential theory and geometric PDE to the nonlinear, fully nonlinear regime.

6. Broader Theoretical and Applied Implications

The subequation subharmonics framework unifies and extends core aspects of analysis, geometry, and PDE theory:

• General Potential Theory: Recovers Laplacian, convex, and plurisubharmonic potential theory as special cases; enables differential inequalities with geometric or convexity constraints as in calibrated geometries.
• Fully Nonlinear PDEs: Provides precise admissibility conditions for viscosity subsolutions, making the theory robust under nonlinear, non-uniformly elliptic, and inhomogeneous operators.
• Singularity and Regularity Theory: Quantifies and stratifies singular sets, suggesting novel strategies for removable singularity results, regularity, and fine structure theorems akin to those for harmonic maps, minimal surfaces, and Yang–Mills fields (Chu, 2016, Harvey et al., 2013).
• Comparison Principles and Uniqueness: Grounded in the geometric structure of the subequation, permitting comparison principles to be proved by geometric, monotonicity-based arguments, relevant for the Dirichlet problem and evolutionary equations.
• Nonlinear Potential Analysis in Geometry: Permits transference of methods and results from classical potential theory to geometric PDEs arising in calibrated, complex, or degenerate geometries, and to nonlinear problems in optimal transport and stochastic processes.

The development and application of subequation subharmonics thus constitute a foundational advance in the analytic and geometric paper of nonlinear elliptic PDEs, allowing the direct application of geometric, topological, and variational techniques in potential theory to high-complexity, nonlinear, and geometric-constraint settings.