Uniform estimates and Brezis-Merle type inequalities for the $k$-Hessian equation

1. Problem Statement and Context

The paper addresses the analysis of the $k$-Hessian equation, a fully nonlinear elliptic partial differential equation defined as:
$$S_k[u] = \sum_{1 \le i_1 < \dots < i_k \le n} \lambda_{i_1} \dots \lambda_{i_k} = f$$
where $\lambda_i$ are the eigenvalues of the Hessian matrix $D^2u$. The operator $S_k$ generalizes the Laplacian ($k=1$) and the Monge-Ampère operator ($k=n$).

The authors focus on $k$-convex functions (functions where the first $k$ elementary symmetric functions of the Hessian eigenvalues are non-negative) vanishing on the boundary of a bounded smooth domain $\Omega \subset \mathbb{R}^n$.

The primary objectives are:

1. To establish Brezis-Merle type exponential integrability inequalities for solutions to the $k$-Hessian equation, particularly in the critical intermediate case where $k = n/2$.
2. To derive Alexandrov-Bakelman-Pucci (ABP) type estimates using these inequalities.
3. To investigate the concentration-compactness principle for the blow-up behavior of solutions to Liouville-type $k$-Hessian equations ($S_k[u] = V e^{-u}$).

2. Methodology

The authors employ a combination of geometric analysis, capacity theory, and iterative techniques:

• Isocapacitary Inequalities: The core of the proof relies on the relationship between the measure of a set and its $k$-Hessian capacity ($Cap_k$). They utilize known isocapacitary inequalities (from XZ14) which bound the volume of a set $E$ in terms of its capacity.
• Comparison Principles: They use a comparison principle for $k$-convex functions (Theorem 2.2) to relate the $k$-Hessian measure of a general function to that of extremal functions associated with capacity.
• Auxiliary Equations and Iteration: For the ABP estimate, the authors construct an auxiliary $k$-Hessian equation involving a specific function $\Phi$. They employ an iterative method inspired by recent work on complex Monge-Ampère equations, utilizing De Giorgi's Lemma to establish uniform bounds.
• Harnack Inequalities: In the study of blow-up behavior, they apply Harnack inequalities for $k$-Hessian equations to control the local behavior of solutions near singular points.

3. Key Contributions and Results

A. Brezis-Merle Type Inequalities (Theorem 1.1)

The paper establishes a fundamental inequality for $k$-convex functions $u \in \mathcal{F}_k(\Omega)$ (vanishing on the boundary) with finite $k$-Hessian mass $M_k(u) = \int_\Omega S_k[u] dx$.

• Subcritical Case ($1 \le k < n/2$): The authors prove $L^p$ estimates for $p$ up to a critical exponent determined by $n$ and $k$.
• Critical Case ($k = n/2$): This is the main novelty. They prove an exponential integrability inequality analogous to the classical Brezis-Merle inequality for the Laplacian in 2D. Specifically, for $0 < \lambda < \alpha_0$ (where $\alpha_0$ is a dimensional constant involving the volume of the unit ball), the following holds:
  $$\sup \left\{ \int_\Omega \exp\left( \lambda \left( \frac{-u}{M_k(u)^{1/k}} \right)^{\frac{k\beta}{k+1}} \right) dx \right\} \le C(n, \lambda, |\Omega|)$$
  This result extends the famous 2D Laplacian result to higher dimensions for the intermediate Hessian operator.

B. ABP Type Estimate (Theorem 1.2)

Using the Brezis-Merle inequality in the critical case ($k=n/2$), the authors derive a refined Alexandrov-Bakelman-Pucci (ABP) estimate.

• Unlike standard ABP estimates which depend on the $L^n$ norm of the right-hand side, this estimate incorporates a logarithmic correction term involving a function $\Phi$.
• The estimate bounds the supremum of the solution $u$ in terms of the boundary values and an integral involving $S_k[-u]$ and $\Phi(\log(S_k[-u]))$. This provides a sharper control on the solution's growth when the source term has specific integrability properties.

C. Concentration-Compactness Principle (Theorem 1.3)

The authors analyze the blow-up behavior of a sequence of solutions $\{u_i\}$ to the Liouville-type equation:
$$S_k[u_i] = V_i e^{-u_i}$$
where $V_i$ are non-negative functions with uniform $L^p$ bounds.

• They define a blow-up set $S$ where $u_i \to -\infty$.
• They prove a trichotomy for a subsequence:
  1. The sequence is locally bounded in $L^\infty$.
  2. The sequence converges to $+\infty$ uniformly on compact subsets.
  3. The blow-up set $S$ is finite and non-empty, and the sequence converges to $+\infty$ uniformly on compact subsets of $\Omega \setminus S$.
• Quantization of Mass: In the third case, the measures $V_i e^{-u_i}$ converge weakly to a sum of Dirac masses $\mu = \sum \alpha_i \delta_{a_i}$. Crucially, the authors prove a quantization condition for the coefficients:
  $$\alpha_i \ge \left( \frac{\alpha_0}{p'} \right)^k$$
  This establishes a lower bound on the "mass" concentration at blow-up points, a hallmark of Brezis-Merle type analysis.

4. Significance and Impact

• Generalization of Classical Results: The paper successfully generalizes the Brezis-Merle theory (originally for the 2D Laplacian) and the Aguilar-Peral theory (for the $n$-Laplacian) to the $k$-Hessian operators in the critical intermediate dimension $k=n/2$.
• New Analytical Tools: The derivation of the ABP estimate using the Brezis-Merle inequality provides a new tool for studying fully nonlinear equations where standard $L^p$ estimates are insufficient.
• Blow-up Analysis: The concentration-compactness result with explicit mass quantization is essential for the existence theory of Liouville-type equations. It helps rule out "vanishing" scenarios and characterizes the singularities of solutions, which is vital for proving existence results via variational methods or degree theory.
• Bridge to Complex Geometry: The techniques and results parallel those found in complex Monge-Ampère geometry, suggesting potential applications in complex differential geometry and Kähler geometry.

In summary, this paper provides a rigorous analytical framework for understanding the regularity, uniform bounds, and singular behavior of solutions to $k$-Hessian equations, filling a significant gap in the theory of fully nonlinear elliptic PDEs in intermediate dimensions.