Harnack Theory and Rigidity for Singular and Degenerate Fully Nonlinear Elliptic Equations with Hamiltonians

Technical Summary of "Harnack Theory and Rigidity for Singular and Degenerate Fully Nonlinear Elliptic Equations with Hamiltonians"

Problem Statement
This paper investigates the regularity, Harnack inequalities, and Liouville-type rigidity properties for viscosity solutions to a broad class of singular or degenerate fully nonlinear elliptic equations. The primary equation under consideration is:
$$\Phi(x, |\nabla u|)F(D^2u) - H(x, \nabla u) + c(x)|u|^{i(\Phi)}u = h(x) \quad \text{in } \Omega,$$
where $\Omega \subset \mathbb{R}^n$ is a bounded domain. The operator involves three critical components:

1. Gradient-dependent ellipticity: The factor $\Phi(x, |\nabla u|)$ captures degeneracy or singularity as the gradient vanishes ($|\nabla u| \to 0$).
2. Fully nonlinear structure: $F$ is a uniformly $(\lambda, \Lambda)$-elliptic operator acting on the Hessian $D^2u$.
3. Hamiltonian term: $H(x, \nabla u)$ introduces a first-order term with specific growth conditions.

The study addresses the Dirichlet problem for this equation, focusing on the interplay between the gradient degeneracy/singularity and the Hamiltonian term, which presents significant challenges for standard regularity and comparison techniques.

Methodology
The authors employ a combination of viscosity solution theory, geometric barrier constructions, and compactness arguments. Key methodological components include:

• Regularity Upgrade: The proof of global $C^{1,\gamma}$-regularity follows a step-by-step upgrade. It begins with the Alexandroff–Bakelman–Pucci (ABP) estimate to establish global boundedness. This is followed by global Lipschitz estimates derived via boundary barriers and a doubling-of-variables argument. Finally, an iterative approximation lemma, driven by compactness arguments and adapted from works by Birindelli, Demengel, Imbert, and Silvestre, is used to establish $C^{1,\gamma}$ regularity.
• Sliding Cusp Method: To establish Harnack inequalities, the authors adapt the "sliding cusp" technique (originally developed by Imbert and Silvestre) rather than the classical sliding paraboloid method. This is necessary because the ellipticity factor $\Phi$ may vanish or become singular when $|\nabla u| \to 0$, a regime where paraboloids (whose gradients vanish at the vertex) fail to maintain contact points away from the singularity. By using a cusp function $\phi(x) = -|x|^{1/2}$, the authors ensure that contact points occur where the gradient is bounded away from zero, allowing for the application of viscosity inequalities.
• Measure Estimates and Ink-Spots: The derivation of Harnack inequalities relies on establishing an $L^\varepsilon$ estimate. This involves proving a measure estimate for supersolutions using the sliding cusp construction and combining it with the "growing ink-spots" lemma (a variant of the Vitali covering lemma) to propagate lower bounds.
• Eigenvalue Theory: The development of a generalized principal eigenvalue theory utilizes Perron's method combined with a refined comparison principle. This requires imposing additional structural assumptions (homogeneity and specific continuity conditions on $\Phi$ and $H$) to overcome the lack of standard comparison principles in the general degenerate/singular setting.

Key Contributions and Results

1. Global $C^{1,\gamma}$-Regularity:
   The paper establishes global $C^{1,\gamma}$-regularity for viscosity solutions to the Dirichlet problem in the absence of zero-order terms ($c(x) \equiv 0$). This result extends previous theories for degenerate/singular equations (e.g., those involving $|\nabla u|^\alpha F(D^2u)$) to include general Hamiltonian terms $H(x, \nabla u)$. The proof handles both the degenerate case ($i(\Phi) \ge 0$) and the singular case ($-1 < i(\Phi) < 0$) by transferring estimates via a transformation that normalizes the operator.

2. Harnack Inequalities:

   • Additive Harnack Inequality: For general operators satisfying assumptions (A1)–(A4), the authors prove an additive Harnack inequality of the form:
     $$\sup_{B_{1/2}} u \le C_H \left( \inf_{B_{1/2}} u + \|h\|_{L^\infty}^{\frac{1}{i(\Phi)+1}} + 1 \right).$$
     The additive constant "+1" is shown to be necessary due to the lack of invariance under amplitude rescaling in the general setting.
   • Standard Harnack Inequality: Under an additional homogeneity assumption (A6), the additive constant is removed, recovering the standard multiplicative Harnack inequality. This leads to the derivation of Liouville-type theorems.

3. Generalized Principal Eigenvalue Theory:
   The paper develops a theory for the Dirichlet principal eigenvalues $\beta^+$ and $\beta^-$ for the operator with a zero-order term. The authors prove the existence of principal eigenfunctions (positive and negative) and relate the eigenvalues to the validity of the maximum and minimum principles. This extends the theory of Birindelli and Demengel to a broader class of operators involving Hamiltonian terms and general gradient-dependent ellipticity factors.

4. Liouville-Type Rigidity Theorems:
   As consequences of the scaled Harnack inequality, the authors prove Liouville-type results for entire solutions in $\mathbb{R}^n$. Specifically, they show that non-negative solutions satisfying certain growth conditions at infinity must be identically zero, and solutions attaining a global minimum must be constant.

Significance and Scope
The paper provides a unified treatment of global regularity and Harnack estimates for singular and degenerate fully nonlinear equations. It encompasses and extends frameworks established in previous works (cited as [7, 8, 24, 56]) by incorporating Hamiltonian terms and a wider class of gradient-dependent ellipticity factors.

The authors claim that their approach to the Harnack inequality via the sliding cusp method is expected to be optimal for the given setting, particularly in handling the contact points where the gradient is small but non-zero. The development of the principal eigenvalue theory for this general class of operators fills a gap in the literature, providing tools to characterize the maximum principle in the presence of both gradient degeneracy and Hamiltonian drift. The results yield new rigidity theorems for entire solutions, contributing to the understanding of the qualitative behavior of solutions to these complex nonlinear equations.