Sigma-2 Hessian Equation

• The sigma-2 Hessian equation is a fully nonlinear elliptic PDE defined by the second elementary symmetric function of Hessian eigenvalues, operating on 2-convex functions.
• It exhibits distinctive boundary blow-up behavior via the Keller–Osserman criterion and relies on refined interior Hessian estimates to achieve higher regularity.
• Recent advances combine viscosity and weak solution theories with convex integration and numerical methods to address challenges in higher-dimensional settings.

Searching arXiv for recent and foundational papers on the sigma-2 Hessian equation to ground the article in the cited literature. The sigma-2 Hessian equation is the fully nonlinear elliptic partial differential equation governed by the second elementary symmetric function of the eigenvalues of the Hessian. For a twice-differentiable function $u$, with Hessian eigenvalues $\lambda_1,\dots,\lambda_n$, the operator is

$$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$

and the model equation is

$$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$

Its ellipticity is restricted to the Gårding cone

$$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$

so the natural admissibility condition is $2$-convexity, namely $\lambda(D^2u(x))\in\Gamma_2$ at every point. The equation occupies an intermediate position between the Poisson equation ($k=1$) and the Monge–Ampère equation ($k=n$), and its modern theory spans boundary blow-up, interior $C^2$ estimates, viscosity regularity, global rigidity, very weak nonuniqueness, and numerical approximation (Covei, 2015).

1. Operator, admissibility, and elliptic structure

The sigma-2 operator is the second elementary symmetric function of the Hessian eigenvalues. In diagonal coordinates it is

$\lambda_1,\dots,\lambda_n$0

and in dimension three it admits the explicit coordinate form

$\lambda_1,\dots,\lambda_n$1

(Shankar et al., 2019). The admissible branch is the Gårding cone

$\lambda_1,\dots,\lambda_n$2

equivalently the class of $\lambda_1,\dots,\lambda_n$3-convex functions. On this cone, the linearization is positive definite. For the constant-right-hand-side equation, the linearized operator is

$\lambda_1,\dots,\lambda_n$4

which is elliptic precisely on $\lambda_1,\dots,\lambda_n$5 (He et al., 2019).

This admissibility restriction is not a peripheral technicality; it is the mechanism that distinguishes the equation from uniformly elliptic models. The operator is elliptic only on a restricted set of functions, and the PDE theory therefore intertwines second-order estimates with convexity-type constraints, viscosity admissibility, and geometric properties of level sets (Froese et al., 2015). In dimension three, one equivalent description of $\lambda_1,\dots,\lambda_n$6 is that the Laplacian restricted to every coordinate plane is nonnegative together with $\lambda_1,\dots,\lambda_n$7, a characterization that is useful both analytically and numerically (Froese et al., 2015).

A recurrent theme in the literature is the distinction between $\lambda_1,\dots,\lambda_n$8-convexity and stricter forms of positivity. One line of work studies classical or viscosity solutions on the admissible branch; another studies semiconvex or dynamically semiconvex regimes, where lower bounds on the minimal eigenvalue relative to $\lambda_1,\dots,\lambda_n$9 or $$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$0 restore enough effective ellipticity for higher-dimensional interior estimates (Shankar et al., 2023).

2. Boundary value problems and Keller–Osserman blow-up theory

A fundamental nonlinear boundary phenomenon is the boundary blow-up problem

$$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$1

posed on a smooth bounded domain $$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$2. For the sigma-2 case, a necessary and sufficient existence criterion is the Keller–Osserman integral condition

$$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$3

under the hypotheses that $$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$4 is $$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$5 on $$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$6, convex, nondecreasing, $$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$7 for $$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$8, $$\sigma_2(D^2u)=\sum_{1\le i<j\le n}\lambda_i\lambda_j =\frac12\Bigl[(\Delta u)^2-|D^2u|^2\Bigr],$$9 for $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$0, and $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$1 is $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$2 and strictly $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$3-convex (Covei, 2015).

The proof uses a subsolution–supersolution framework. A strict subsolution $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$4 is obtained from a Dirichlet problem with constant boundary value $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$5 and right-hand side $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$6, exploiting monotonicity of $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$7. A supersolution $$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$8 is built from the large-boundary-data linear problem

$$\sigma_2(D^2u)=1 \quad\text{or more generally}\quad \sigma_2(D^2u)=f(x,u,Du).$$9

together with Maclaurin’s inequality $$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$0. Solving finite Dirichlet problems

$$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$1

and passing to the monotone limit yields a blow-up solution between the two barriers (Covei, 2015).

Two model nonlinearities exhibit the sharpness of the criterion. For $$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$2, the Keller–Osserman integral converges exactly when $$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$3, so existence of a positive boundary-blow-up solution is equivalent to $$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$4. For $$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$5, the integral always converges, hence the equation

$$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$6

admits at least one positive blow-up solution (Covei, 2015).

This theory shows that the sigma-2 equation shares the classical Keller–Osserman dichotomy but with the exponent $$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$7 in the integral condition, reflecting the algebraic structure of the second Hessian operator rather than the Laplacian or Monge–Ampère cases. A plausible implication is that boundary singularity formation for $$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$8 is already rigidly encoded at the level of one-dimensional integral growth conditions.

3. Interior Hessian estimates and regularity theory

Interior $$\Gamma_2=\{\lambda\in\mathbb R^n:\sigma_1(\lambda)>0,\ \sigma_2(\lambda)>0\},$$9 estimates are central because once $2$0 is controlled, the equation becomes uniformly elliptic and concave in the admissible cone, so Evans–Krylov and Schauder theory yield higher regularity. In dimension three, a priori interior $2$1 bounds for

$2$2

were established using a combination of differentiated identities, a mean-value inequality on a graph submanifold, and a maximum-principle argument for a refined scalar test function (Qiu, 2017).

For semiconvex solutions of the constant equation

$2$3

there is an interior Hessian estimate of exponential type: $2$4 assuming $2$5 and $2$6 (Shankar et al., 2019). An equivalent version uses $2$7. The proof rests on three ingredients: a Jacobi-type differential inequality for $2$8, the Legendre–Löwy transform

$2$9

which converts the nonuniformly elliptic linearization into a uniformly elliptic equation for $\lambda(D^2u(x))\in\Gamma_2$0, and a mean-value inequality for the transformed operator (Shankar et al., 2019). The novelty is that semiconvexity $\lambda(D^2u(x))\in\Gamma_2$1 with arbitrary $\lambda(D^2u(x))\in\Gamma_2$2 suffices, improving earlier “almost convex” hypotheses.

Dimension four marks a threshold in the known a priori theory. For admissible $\lambda(D^2u(x))\in\Gamma_2$3 solving

$\lambda(D^2u(x))\in\Gamma_2$4

with $\lambda(D^2u(x))\in\Gamma_2$5 and $\lambda(D^2u(x))\in\Gamma_2$6, there is an interior estimate

$\lambda(D^2u(x))\in\Gamma_2$7

(Fan, 3 Sep 2025). The proof combines an almost-Jacobi inequality for $\lambda(D^2u(x))\in\Gamma_2$8, a Pogorelov-type doubling argument based on a test function of the form

$\lambda(D^2u(x))\in\Gamma_2$9

and a small-scale control obtained from almost-everywhere twice differentiability of Lipschitz viscosity solutions together with a generalized Savin small-perturbation theorem (Fan, 3 Sep 2025). A closely related four-dimensional estimate for $k=1$0 and then $k=1$1 was established earlier by a doubling-plus-blow-up method (Shankar et al., 2023).

In dimensions $k=1$2, the general interior regularity question remains open, and Pogorelov-type examples obstruct dimension-free estimates for all admissible solutions. Existing higher-dimensional $k=1$3 bounds require extra dynamic semi-convexity conditions, such as a lower bound on $k=1$4 in terms of $k=1$5 or $k=1$6 (Shankar et al., 2023). This suggests that the borderline between regularity and singularity in higher dimensions is not merely technical but structural.

Recent work adds an a priori interior $k=1$7 estimate in terms of the $k=1$8 norm, for any $k=1$9: $k=n$0 for smooth solutions of $k=n$1 in $k=n$2 (Mooney, 20 May 2025). The proof rewrites the linearized equation in divergence form as a Laplace–Beltrami equation, uses Krylov–Safonov and ABP on the sublevel sets of $k=n$3, and then exploits the algebraic relation $k=n$4 to bound all eigenvalues once $k=n$5 is controlled (Mooney, 20 May 2025).

4. Convexity, strict $k=n$6-convexity, and geometric interpretation

The sigma-2 equation has a strong geometric side. For convex viscosity solutions of

$k=n$7

strict $k=n$8-convexity means that every supporting linear function has a contact set of dimension at most $k=n$9. This property was proved by constructing paraboloid barriers in thin cylinders and showing that any $C^2$0-dimensional contact set would contradict the viscosity supersolution condition (Mooney, 2020).

The same work derives short proofs of smoothness and interior $C^2$1 estimates for convex viscosity solutions of

$C^2$2

yielding

$C^2$3

and then full interior $C^2$4 regularity by Evans–Krylov and Schauder bootstrap (Mooney, 2020). The argument passes from qualitative strict $C^2$5-convexity to a quantitative control on the geometry of sublevel sets and then invokes classical Dirichlet solvability and Pogorelov-type interior estimates.

A more geometric strengthening states that viscosity solutions of

$C^2$6

cannot touch a harmonic function on a smooth embedded piece of a minimal hypersurface from below (Mooney, 20 May 2025). The barrier used there has the form

$C^2$7

where $C^2$8 is signed distance to the minimal surface, $C^2$9 is nearest-point projection, and $\lambda_1,\dots,\lambda_n$00. The non-contact theorem can be viewed as a form of strict $\lambda_1,\dots,\lambda_n$01-convexity and, in the convex setting, forces the contact set with any supporting hyperplane to have dimension at most $\lambda_1,\dots,\lambda_n$02 (Mooney, 20 May 2025).

The equation also connects to special Lagrangian geometry. In dimension three, the critical-phase special Lagrangian equation

$\lambda_1,\dots,\lambda_n$03

is equivalent to $\lambda_1,\dots,\lambda_n$04, so rigidity for sigma-2 solutions implies a Bernstein-type theorem for global special Lagrangian graphs in $\lambda_1,\dots,\lambda_n$05 under quadratic growth (Chen et al., 2018). More generally, the reformulation of $\lambda_1,\dots,\lambda_n$06 as a twisted special Lagrangian equation in a weighted Euclidean metric yields interior $\lambda_1,\dots,\lambda_n$07 regularity for continuous viscosity solutions in dimension three when $\lambda_1,\dots,\lambda_n$08 is positive and Lipschitz (Zhou, 2023).

These results show that “strict $\lambda_1,\dots,\lambda_n$09-convexity” is not merely an eigenvalue condition. It manifests through contact geometry, sublevel-set thickness, non-contact with minimal-surface harmonic data, and the geometry of gradient graphs.

5. Entire solutions, Liouville theorems, and rigidity

A major global theme is rigidity of entire admissible solutions. If $\lambda_1,\dots,\lambda_n$10, $\lambda_1,\dots,\lambda_n$11, is an entire $\lambda_1,\dots,\lambda_n$12-convex solution of

$\lambda_1,\dots,\lambda_n$13

with quadratic growth from below,

$\lambda_1,\dots,\lambda_n$14

then $\lambda_1,\dots,\lambda_n$15 must be a quadratic polynomial (He et al., 2019). The proof rescales $\lambda_1,\dots,\lambda_n$16 on sublevel sets $\lambda_1,\dots,\lambda_n$17, uses a global gradient estimate on mean-convex domains together with a Pogorelov estimate, and obtains a uniform bound on $\lambda_1,\dots,\lambda_n$18. Evans–Krylov theory then implies higher regularity, and the limit argument forces $\lambda_1,\dots,\lambda_n$19 to be constant (He et al., 2019).

A related rigidity theorem holds in all dimensions for entire $\lambda_1,\dots,\lambda_n$20-convex solutions of $\lambda_1,\dots,\lambda_n$21 that satisfy quadratic growth and a one-sided bound

$\lambda_1,\dots,\lambda_n$22

Under these assumptions, the solution is again a quadratic polynomial; in dimension three the $\lambda_1,\dots,\lambda_n$23-assumption is redundant (Chen et al., 2018). The proof proceeds through a local Pogorelov-type estimate in bounded domains,

$\lambda_1,\dots,\lambda_n$24

followed by scaling and then Evans–Krylov and Schauder theory (Chen et al., 2018).

The literature frames these results as an extension of Jörgens–Calabi–Pogorelov-type rigidity from the Monge–Ampère equation to the intermediate Hessian setting (He et al., 2019). A plausible implication is that, for admissible entire sigma-2 solutions, the main obstruction to nonquadratic behavior is not lack of global convexity alone but the combination of growth, admissibility, and interior $\lambda_1,\dots,\lambda_n$25 control.

Higher-dimensional regularity results also feed into Liouville theory. In four dimensions, interior Hessian estimates imply that $\lambda_1,\dots,\lambda_n$26-viscosity solutions are $\lambda_1,\dots,\lambda_n$27 in the interior, and entire solutions with quadratic growth under the dynamic semi-convexity condition are necessarily quadratic polynomials (Shankar et al., 2023).

6. Weak solution theories, nonuniqueness, and computational methods

The sigma-2 equation supports several inequivalent weak frameworks. The viscosity theory is stable under uniform limits and coincides with classical theory on smooth $\lambda_1,\dots,\lambda_n$28-convex solutions (Mooney, 20 May 2025). By contrast, “very weak solutions” may be defined through a double-divergence identity: $\lambda_1,\dots,\lambda_n$29 for all $\lambda_1,\dots,\lambda_n$30, where $\lambda_1,\dots,\lambda_n$31 (Li et al., 2024). Using convex integration and a cut-off technique, it was shown that for $\lambda_1,\dots,\lambda_n$32 there exist infinitely many $\lambda_1,\dots,\lambda_n$33 very weak solutions of the Dirichlet problem with prescribed boundary value, under $\lambda_1,\dots,\lambda_n$34, $\lambda_1,\dots,\lambda_n$35, and $\lambda_1,\dots,\lambda_n$36 (Li et al., 2024).

This establishes low-regularity flexibility and nonuniqueness in sharp contrast with the classical viscosity theory, where smooth or admissible solutions are unique under the usual structural assumptions (Li et al., 2024). The contrast is substantive rather than terminological: the convex integration solutions are very weak solutions, not viscosity solutions. This addresses a common misconception that all reasonable weak formulations of $\lambda_1,\dots,\lambda_n$37 should share the same uniqueness and regularity properties.

The computational literature reflects the same admissibility issues. A wide-stencil monotone finite-difference scheme for the three-dimensional Dirichlet problem is provably convergent to the viscosity solution via the Barles–Souganidis framework, while a standard centered-difference scheme is more accurate but not monotone and thus lacks an unconditional convergence proof (Froese et al., 2015). Both are typically solved with Newton’s method. For smooth solutions, the accurate scheme exhibits second-order behavior in experiments, whereas the monotone scheme is usually first order unless very wide stencils are used (Froese et al., 2015).

A nine-point finite-difference discretization for $\lambda_1,\dots,\lambda_n$38-Hessian equations yields a locally unique discrete solution with $\lambda_1,\dots,\lambda_n$39 error for smooth nondegenerate solutions, and Newton’s method converges quadratically once iterates enter the local contraction ball (Awanou, 2014). That work also introduced parameter-free subharmonicity-preserving and nonlinear Gauss–Seidel-type iterations for the $\lambda_1,\dots,\lambda_n$40-Hessian case, designed to preserve discrete $\lambda_1,\dots,\lambda_n$41-convexity and to handle non-smooth solutions in practice (Awanou, 2014).

7. Dimension dependence, extensions, and open directions

The theory of the sigma-2 Hessian equation is sharply dimension-dependent. In dimensions $\lambda_1,\dots,\lambda_n$42, the available interior Hessian estimates are strong enough to force smoothness of viscosity solutions in broad regimes; in higher dimensions the general interior regularity problem remains open (Mooney, 20 May 2025). Dimension four is a particularly important threshold: the constant-right-hand-side equation and positive $\lambda_1,\dots,\lambda_n$43 inhomogeneous equations admit interior Hessian estimates without extra structural assumptions there, while in $\lambda_1,\dots,\lambda_n$44 one must impose dynamic semi-convexity or related lower bounds on the smallest eigenvalue (Fan, 3 Sep 2025).

Several extensions move beyond the pure equation $\lambda_1,\dots,\lambda_n$45. In dimension three, there are interior $\lambda_1,\dots,\lambda_n$46 estimates for $\lambda_1,\dots,\lambda_n$47-viscosity solutions of

$\lambda_1,\dots,\lambda_n$48

with positive Lipschitz $\lambda_1,\dots,\lambda_n$49, obtained through the twisted special Lagrangian formulation (Zhou, 2023). More recent work establishes interior Hessian estimates for

$\lambda_1,\dots,\lambda_n$50

in dimension three for $\lambda_1,\dots,\lambda_n$51-convex solutions with $\lambda_1,\dots,\lambda_n$52 bounds, and in higher dimensions under semiconvexity assumptions (Jiao et al., 15 Feb 2026). In dimension four, positive $\lambda_1,\dots,\lambda_n$53 right-hand sides $\lambda_1,\dots,\lambda_n$54 can also be treated (Fan, 3 Sep 2025).

Another active direction concerns Pogorelov-type $\lambda_1,\dots,\lambda_n$55 estimates. For admissible solutions of the Dirichlet problem

$\lambda_1,\dots,\lambda_n$56

with $\lambda_1,\dots,\lambda_n$57, $\lambda_1,\dots,\lambda_n$58, and a uniform lower bound $\lambda_1,\dots,\lambda_n$59, one has an interior estimate

$\lambda_1,\dots,\lambda_n$60

and a Pogorelov bound

$\lambda_1,\dots,\lambda_n$61

(Li et al., 9 Apr 2025). The dependence on the lower bound $\lambda_1,\dots,\lambda_n$62 reflects the intermediate-Hessian nature of the equation; the paper notes that this hypothesis can be removed when $\lambda_1,\dots,\lambda_n$63, but not for $\lambda_1,\dots,\lambda_n$64 (Li et al., 9 Apr 2025).

Across these developments, a consistent pattern emerges. The sigma-2 Hessian equation is governed by a tension between ellipticity restricted to $\lambda_1,\dots,\lambda_n$65, geometric strengthening such as strict $\lambda_1,\dots,\lambda_n$66-convexity, and the search for dimension-sensitive mechanisms that restore effective uniform ellipticity. This suggests that future progress will likely continue to depend on hybrid tools: Jacobi inequalities, convexity transforms, doubling arguments, small-perturbation theory, and geometric barrier constructions rather than a single universal regularity principle.