Fully Nonlinear Uniformly Elliptic Equations

Updated 8 December 2025

Fully nonlinear uniformly elliptic equations are second-order PDEs defined by a nonlinear Hessian dependency and controlled by Pucci operators ensuring strict positivity and boundedness.
The viscosity solution framework leverages ABP estimates, Harnack inequalities, and the Evans–Krylov theorem to establish regularity, existence, and uniqueness.
These equations have wide applications in optimal control, geometric analysis, and stochastic homogenization, with quantitative error bounds and convergence results guided by uniform ellipticity.

A fully nonlinear uniformly elliptic equation is a second-order partial differential equation (PDE) in non-divergence form in which the principal part is a nonlinear function of the Hessian and, possibly, lower-order terms, with strict positivity and boundedness imposed on the linearization. These equations underlie many models in nonlinear analysis, stochastic control, geometry, and applied PDE, with regularity, existence, uniqueness, and qualitative properties determined by the uniform ellipticity structure.

1. Definition, Canonical Examples, and Uniform Ellipticity

The general form of a fully nonlinear uniformly elliptic equation on an open set $U\subset \mathbb{R}^d$ is

$F(D^2u(x), Du(x), u(x), x) = 0, \quad x\in U.$

The operator $F: \mathbb{S}^d \times \mathbb{R}^d \times \mathbb{R} \times U \to \mathbb{R}$ is continuous, possibly only measurable in $x$ , and "fully nonlinear" in that the dependence on $D^2 u$ is arbitrary, not necessarily linear or affine.

Uniform ellipticity is formulated via the Pucci extremal operators. For symmetric matrices $M, N$ ,

$\mathcal{M}^-_{\lambda,\Lambda}(M-N) \le F(M, p, s, x) - F(N, p, s, x) \le \mathcal{M}^+_{\lambda,\Lambda}(M-N),$

where, for $e_i$ the eigenvalues of $M$ ,

$\mathcal{M}^+_{\lambda,\Lambda}(M) = \Lambda \sum_{e_i>0} e_i - \lambda \sum_{e_i<0} e_i, \quad \mathcal{M}^-_{\lambda,\Lambda}(M) = \lambda \sum_{e_i>0} e_i - \Lambda \sum_{e_i<0} e_i.$

Typical models include Bellman–Isaacs equations (optimal control/game theory), Monge–Ampère (geometry, optimal transport) and Hamilton–Jacobi–Bellman equations (path-dependent control). The uniform ellipticity constants $0 < \lambda \le \Lambda < \infty$ control the degeneracy permitted in $D^2u$ and are central to all core estimates (Chang-Lara, 26 Jan 2024, Armstrong et al., 2012, Zhang, 27 Nov 2025).

2. Viscosity Solutions and Regularity Theory

The viscosity solution paradigm furnishes a well-posedness framework for fully nonlinear equations lacking variational structure or classical weak derivative interpretations. A continuous $u$ is a viscosity subsolution of $F(D^2u, \cdots) = 0$ in $U$ if, when a smooth function $\varphi$ touches $u$ from above at $x_0$ ,

$F(D^2\varphi(x_0), \cdots) \le 0;$

a viscosity supersolution if, under touching from below, $F(D^2\varphi(x_0), \cdots) \ge 0$ .

Uniform ellipticity alone ensures robust comparison principles, compactness under limits, and existence via Perron's method. The key technique in the regularity theory is the use of ABP-type (Aleksandrov–Bakelman–Pucci) measure estimates, Harnack inequalities, and barrier arguments (Chang-Lara, 26 Jan 2024, Armstrong et al., 2011, Silvestre et al., 2013). The Caffarelli–Krylov–Safonov theory provides interior $C^{0,\alpha}$ and $C^{1,\alpha}$ regularity, while the Evans–Krylov theorem yields $C^{2,\alpha}$ regularity for convex/concave $F$ .

3. Quantitative and Qualitative Regularity Results

Interior Regularity. For $F$ convex/concave in $M$ , $F(D^2u)=f(x)$ with $f\in C^\alpha$ , the Evans–Krylov theorem gives

$\|u\|_{C^{2,\alpha}(B_{1/2})} \le C(\|u\|_{L^\infty(B_1)} + \|f\|_{C^\alpha(B_1)}).$

In general, optimal $W^{2,\varepsilon}$ regularity up to a closed singular set of dimension $n-\varepsilon$ is achieved for arbitrary uniformly elliptic $F$ , with $\varepsilon$ depending polynomially on $\lambda/\Lambda$ (Le, 2018, Armstrong et al., 2011).

Fractional Sobolev Regularity. For $F$ (possibly nonconvex) uniformly elliptic and $u$ a viscosity solution of $F(D^2u)=f\in L^p$ , $p > n$ , there exists $\varepsilon = \varepsilon(n, \lambda, \Lambda)$ such that $u \in W^{1+\varepsilon,p}$ , equivalently $(-\Delta)^\theta u\in L^p$ for $\frac{1}{2}<\theta<1$ (Pimentel et al., 2022).
Boundary Regularity. Viscosity solutions of $F(D^2u, Du, x)=f$ in $C^2$ domains with appropriate data satisfy boundary $C^{2,\alpha}$ expansion, with the coefficients determined by the equation evaluated at the boundary (Silvestre et al., 2013).
Dimension Two. In $n=2$ dimensions, full interior $C^{2,\alpha}$ regularity is available for general (nonconvex) $F$ , with exponent and constants depending only on the ellipticity ratio (Zhang, 27 Nov 2025).

4. Partial Regularity and Counterexamples

For general (nonconvex) $F$ , optimal partial regularity holds: viscosity solutions are $C^{2,\alpha}$ off a closed singular set whose Hausdorff dimension is strictly less than $n$ , with upper bound $n-\varepsilon$ governed by the ellipticity ratio (Armstrong et al., 2011). For $F$ convex/concave, the singular set is empty.

Sharp examples show that as $\Lambda/\lambda \to \infty$ , $\varepsilon \to 0$ and full $C^{2,\alpha}$ regularity cannot be expected without additional structure. Notably, counterexamples constructed by Nadirashvili–Vļăduţ demonstrate that nonconvexity (even with smooth $F$ ) may yield non- $C^2$ solutions.

5. Existence, Uniqueness, and Approximation Methods

Existence and Uniqueness. Under uniform ellipticity and Lipschitz dependence, viscosity solutions to the Dirichlet problem exist and are unique, with uniqueness following from the comparison principle (Krylov, 2012, Ren, 2014).
Approximation and Error Estimates. For equations with discontinuous (measurable) coefficients, solutions can be approximated by solutions to convex–capped or mollified equations, with quantitative $O(\delta^\alpha)$ error estimates in terms of the discretization or regularization parameter (Turanova, 2013, Krylov, 2012).

For monotone, consistent, finite-difference schemes, the convergence rate of discrete approximations to viscosity solutions is controlled by the same structural parameters and matches the continuum error bounds.

6. Advanced Topics and Applications

Stochastic Homogenization. In random environments, Armstrong–Smart establish almost sure homogenization of solutions to random, stationary-ergodic, uniformly elliptic equations. The obstacle problem approach and subadditive ergodic theorem yield the effective (homogenized) operator $\overline F$ with the same uniform ellipticity constants (Armstrong et al., 2012).
Unique Continuation. For $C^{1,1}$ uniformly elliptic $F$ , viscosity solutions vanishing on an open set (or at a point of infinite order) must vanish identically. The proof combines boundary Harnack inequalities, Savin regularity ("flat–implies– $C^{2,\alpha}$ "), and linear unique continuation (Armstrong et al., 2011).
Liouville Theorems and Classification in Halfspaces. All nonnegative viscosity solutions to $F(D^2u)=0$ in $\mathbb{R}^n_+$ with zero boundary data are necessarily affine in $x_n$ ; with $F(D^2u)=1$ they are quadratic polynomials in $x_n$ (Lian, 20 Nov 2025, Leoni, 2011). Power-type nonlinearities are classified via critical exponents determined by ellipticity ratio.
Degenerate/Singular Equations with Lower-order Terms. Equations with $L_d$ -drift and BMO coefficients admit interior $W^2_p$ a priori estimates for $p>d_0(d,\lambda,\|b\|_{L_d})>d/2$ , with global existence in $W^2_p$ for data in $L_p$ , even in the absence of convexity/concavity (Krylov, 2020).
Quadratic Gradient Growth and Multiplicity. Fully nonlinear equations with natural quadratic growth in $Du$ demonstrate existence, compactness, and multiplicity by combining boundary half-Harnack inequalities, Vázquez-type strong maximum principles, and topological degree theory (Nornberg et al., 2018).

7. Partial Uniform Ellipticity, Level-Set Methods, and Extensions

Yuan develops a level-set approach to "partial uniform ellipticity," establishing that many structural consequences—particularly second-order a priori bounds, Harnack inequalities, and Evans–Krylov-type interior regularity—follow from weaker slice-wise convexity hypotheses, enabling application to Hessian equations and parabolic flows where full uniform ellipticity fails (Yuan, 2022).

Applications span stochastic games, path-dependent PDEs, nonlocal Isaacs–Lévy PDEs, and problems with $L^p$ or $L^d$ drift. The viscosity solution and uniform ellipticity framework has become standard across nonlinear analysis, stochastic homogenization, and geometric PDEs.