--- title: Nonlinear Potential Theory url: https://www.emergentmind.com/topics/nonlinear-potential-theory type: topic --- # Nonlinear Potential Theory Nonlinear potential theory is a unified framework for analyzing subharmonic functions and associated subsolutions of degenerate elliptic and parabolic partial differential equations (PDEs), driven by the interaction between geometric constraint sets in the jet bundle and fully nonlinear operator theory. The modern approach, developed notably by Harvey and Lawson and their collaborators, systematically organizes the classical, quasilinear, and fully nonlinear cases by encoding operator properties in closed subequation sets on 2-jets, leading to powerful comparison principles, correspondence theorems, and geometric applications extending well beyond classical viscosity methods [2303.16735]. ## 1. Subequations, Nonlinear Subharmonics, and the Jet Bundle Let $X \subset \mathbb{R}^n$ be open and consider the 2-jet bundle $J^2(X) = X \times \mathbb{R} \times \mathbb{R}^n \times S(n)$, encoding point, value, gradient, and Hessian. A **subequation** $F \subset J^2(X)$ is a closed subset subject to the following axioms: - **Positivity (degenerate ellipticity):** For all $x \in X$, $(r, p, A) \in F_x$ implies $(r, p, A+P) \in F_x$ for all $P \geq 0$. - **Negativity (properness):** $(r, p, A) \in F_x$ implies $(r-s,p,A) \in F_x$ for all $s>0$. - **Topological stability:** $F$ is closed in $J^2(X)$ and, writing $F_x := F \cap (\{x\}\times J^2)$, one has $F = \operatorname{Int} F$, $F_x = \operatorname{Int}(F_x)$, $(\operatorname{Int} F)_x = \operatorname{Int}(F_x)$. These properties ensure that $F$ encodes the positivity and properness (degenerate ellipticity) of the associated operator. An upper semicontinuous function $u$ is called **$F$-subharmonic** if, for every $x_0 \in X$ and every upper test jet $(r, p, A)$ of $u$ at $x_0$, one has $(x_0, r, p, A) \in F$. The **Dirichlet dual** is defined by $\widetilde{F} = -[\operatorname{Int} F]^c$, and $u$ is $F$-superharmonic if and only if $-u$ is $\widetilde{F}$-subharmonic. This jet-based formulation generalizes the notion of subharmonic functions and solutions to fully nonlinear PDEs, and encompasses classical linear, quasilinear ($p$-Laplacian), and complex/real pluripotential theories [2303.16735, 2203.14015, 2303.14477]. ## 2. Monotonicity Cones, Duality, and the Comparison Principle A central structural device is the **monotonicity cone** $M \subset J^2$ (constant-coefficient, closed, convex, vertex at $0$), which itself satisfies the subequation axioms. $F$ is said to be **$M$-monotone** if $F_x + M \subset F_x$ for all $x$. This monotonicity allows for: - The jet addition identity $F_x + \widetilde{F}_x \subset M_x$. - Comparison between subharmonic-superharmonic pairs via the **zero-maximum principle** for $M$, a generalized maximum principle for functions in the $M$-subharmonic class. The minimal monotonicity cone is $M_0 = \{(s, 0, P): s \leq 0, P \geq 0\}$, automatically associated with degenerate ellipticity and properness, but strictly stronger monotonicity is often imposed for finer comparison results [2303.16735, 2009.01611]. Given an $M$ (with $\operatorname{Int} M \neq \varnothing$) admitting a strictly $M$-subharmonic $C^2$ function $\psi$ on a bounded domain $\Omega$, one has the **zero-maximum principle (ZMP):** Any upper semicontinuous $z$ with $z \in M(\Omega)$ and $z \leq 0$ on $\partial \Omega$ satisfies $z \leq 0$ in $\Omega$. Through monotonicity and duality, this yields the **general comparison principle:** If $F$ is fiberegular and $M$-monotone, then for $u \in \mathrm{USC}(\Omega)$ $F$-subharmonic and $w \in \mathrm{LSC}(\Omega)$ $F$-superharmonic, $u \leq w$ on $\partial\Omega$ implies $u \leq w$ on $\Omega$ (see also parabolic and reduced-boundary versions) [2303.16735, 2203.14015]. ## 3. Fiberegularity and Variable-Coefficient Settings **Fiberegularity** is the condition that the fiber map $\mathcal{F}: X \to$ closed subsets of $J^2$, $x \mapsto F_x$ is continuous in the Hausdorff metric. This ensures that local structural features (e.g., monotonicity and barrier arguments) can be transferred from the constant-coefficient to variable-coefficient settings, underpinning comparison proofs via convolution and localization methods. If $M_x + F_x \subset F_x$ and $F$ is fiberegular, one can guarantee comparison and regularity results under only mild continuity in the coefficients of the operator. Fiberegularity is precisely what enables the generalized comparison arguments for non-constant settings required by complex and degenerate geometries [2303.16735, 2203.14015]. ## 4. Correspondence Principle: Relating Subequations and Fully Nonlinear PDEs Nonlinear potential theory is not just an abstract formulation; it provides a **correspondence principle** linking subequation classes with (viscosity) subsolutions of fully nonlinear second-order elliptic PDEs \[ F(x, u, Du, D^2u) = 0. \] An operator–subequation pair $(F, G)$ is called proper-elliptic if $G \subset J^2(X)$ is a subequation, $F \in C(G)$ satisfies \[ F(x,r,p,A) \leq F(x,r-s,p,A+P) \quad \forall s > 0, P \geq 0, \] and admissible subsolutions are defined by the jet constraint $J \in G_x$, $F(x, J) \geq 0$ at each test jet. The induced subequation $\mathcal{F} := \{(x,J) \in G: F(x, J) > 0\}$ is fiberegular and $M$-monotone (provided $G$ and $F$ have mild structural continuity). Then upper semicontinuous $u$ is $\mathcal{F}$-subharmonic on $X$ if and only if it is a $G$-admissible viscosity subsolution of $F=0$. Thus, comparison principles for $\mathcal{F}$-subharmonics immediately yield comparison for (possibly constrained) viscosity solutions of $F$. This compatibility encompasses operator classes (e.g., degenerate elliptic, gradient-dependent, Dirichlet–Gårding, transport, weakly parabolic) that go well beyond the reach of classical viscosity theory [2303.16735, 2203.14015, 2009.01611]. ## 5. Canonical Classes and Model Examples Nonlinear potential theory organizes a broad family of constrained/unconstrained PDEs as instances of jet-based subequation frameworks. Key model examples include: - **Optimal transport operators:** $g(Du)\det(D^2u) = f(x)$, with $g$ strictly increasing in a cone $D$; the monotonicity cone $M(D,P) = \{p \in D, A \geq 0\}$. - **Hyperbolic polynomial models:** $g(Du) = 0$ with $g$ homogeneous and hyperbolic in the sense of Gårding (e.g., $g(p_1, p_2) = p_1^2 - p_2^2$ and $D = \{p_1 > |p_2|\}$); monotonicity via a directional cone $M(D)$. - **Nonstandard Monge–Ampère perturbations:** $F(x, p, A) = \det(A + M(x, p)) - f(x)$, where $M(x, p)$ depends linearly on $p$ via continuous (possibly non-Lipschitz) maps; monotonicity via a half-space in $p$. - **Weakly parabolic operators:** The comparison and correspondence extend under fiberegularity to parabolic structures (e.g., Krylov-type monotonicity). In each case, the subequation structure plus sufficient monotonicity and fiberegularity allow for comparison theorems and well-posedness even when standard viscosity structural hypotheses (Crandall–Ishii–Lions) fail [2303.16735]. ### Table: Model Operators in Nonlinear Potential Theory | Operator Type | Subequation Structure & Monotonicity | Key Features | |-------------------------------|------------------------------------------------|-----------------------------------| | $g(Du)\det(D^2u) = f(x)$ | $M(D,P)$ monotonicity, $D$-increasing $g$ | Optimal transport, fully nonlinear| | Hyperbolic polynomials | Directional monotonicity cone $M(D)$ | Gårding hyperbolic, directionality| | $\det(A + M(x, p)) - f(x)$ | Cone via $D = \{(b(x), q) \geq 0\}$ | Gradient dependence, no Lipschitz | | Weakly parabolic (Krylov) | Parabolic monotonicity, reduced boundary | Comparison beyond ellipticity | ## 6. Applications and Extensions The nonlinear potential theory framework enables: - Geometric proofs of inequalities (e.g., Penrose, Minkowski) via monotonicity formulas for $p$-capacitary potentials [2205.11642, 1906.00322]. - Regularity, capacity, and comparison results for quasilinear and fully nonlinear PDEs in domains, including boundary value problems under minimal regularity [1304.1312, 2510.13269]. - Unified treatment of both degenerate elliptic and weak parabolic problems, and geometric flows via jet-based monotonicity and duality principles. - Extensions to a general class of constraint sets relevant in complex geometry (e.g., Kähler, calibrated geometries, Hessian equations) [2203.14015]. - Robust handling of variable-coefficient and nonlocal structures through fiberegularity, with applications to mixed local/nonlocal equations [2510.13269, 2402.04809]. The theory circumvents classical regularity and structure requirements—such as Lipschitz continuity in matrix coefficients—replacing these with geometric monotonicity and fiber continuity. In particular, the monotonicity–duality method proves decisive in domains and for operators where viscosity structure theorems are inapplicable [2303.16735, 2203.14015]. ## 7. Broader Context and Open Questions Nonlinear potential theory has reshaped the analysis of degenerate and fully nonlinear equations by encoding operator data in jet geometry and enabling powerful comparison and correspondence theorems. Open directions include: - Classification of subequations admitting canonical operators. - Sharper regularity theory for general $F$-harmonics. - Extension to PDEs with weaker monotonicity or with measure data. - Dirichlet problems for singular or nonpseudoconvex boundaries. - Geometric applications to isoperimetric, conformal, and calibrated manifolds. The subequation approach, founded on monotonicity, duality, and fiberegularity, provides a flexible, geometrically transparent, and highly general framework for the theory and applications of nonlinear PDEs [2303.16735, 2203.14015]. Source: https://www.emergentmind.com/topics/nonlinear-potential-theory