ABP estimate and Harnack inequality for a class of degenerate fully nonlinear pseudo-$p$-Laplacian equations

Technical Summary: ABP Estimate and Harnack Inequality for a Class of Degenerate Fully Nonlinear Pseudo-p-Laplacian Equations

Problem Statement
This paper investigates the basic regularity theory for viscosity solutions of a specific class of degenerate fully nonlinear pseudo-$p$-Laplacian equations in nondivergence form. The primary object of study is the inequality system:
$$\begin{cases} M^{-}_{\lambda,\Lambda}\left(\left[|D_i u|^{p/2}\right] D^2 u \left[|D_i u|^{p/2}\right]\right) - \Lambda|Du|^{p+1} \leq f(x), \\ M^{+}_{\lambda,\Lambda}\left(\left[|D_i u|^{p/2}\right] D^2 u \left[|D_i u|^{p/2}\right]\right) + \Lambda|Du|^{p+1} \geq f(x), \end{cases}$$
defined in the unit ball $B_1$, where $p \geq 0$, $0 < \lambda \leq \Lambda$, $f \in L^n(B_1)$, and $[\cdot]$ denotes a diagonal matrix.

The central difficulty addressed is the coordinatewise degeneracy inherent in these operators. Unlike the standard $p$-Laplacian, which degenerates only when the full gradient $|Du|$ vanishes, the pseudo-$p$-Laplacian loses ellipticity whenever any single partial derivative $D_j u$ vanishes. This creates a degeneracy set much larger than $\{|Du|=0\}$. Furthermore, at points where $D_j u = 0$, the second derivative $D_{jj}u$ may become unbounded, while derivatives in non-degenerate directions remain controlled. This anisotropy prevents the direct application of standard regularity results (such as those by Imbert-Silvestre or Mooney) which rely on uniform ellipticity or degeneracy only where the gradient is large.

Methodology
The authors develop a regularity theory adapted to this specific anisotropic degeneracy through two main technical innovations:

1. Adapted Aleksandrov-Bakelman-Pucci (ABP) Estimate:
   Instead of mapping the standard gradient $Du$, the proof utilizes the area formula on the map $G(z) = (|z_i|^p z_i)$. This choice is motivated by the structure of the operator, effectively estimating the image of $(|D_i u|^p D_i u)$ rather than $Du$. This allows the authors to handle the specific scaling properties of the pseudo-$p$-Laplacian.

2. Sliding Paraboloid with Anisotropic Functions:
   To establish the Harnack inequality, the authors adapt the sliding paraboloid method (originally by Caffarelli and Savin). Standard paraboloids ($|x|^2$) fail to handle the coordinatewise degeneracy. Instead, the authors introduce an anisotropic "paraboloid" function:
   $$\phi(x) = -1 + \frac{p}{2+p} \left( |x_1|^{1+\frac{1}{1+p}} + \cdots + |x_n|^{1+\frac{1}{1+p}} \right).$$
   This function satisfies a crucial identity: $-(1+p)[|D_i \phi|^{p/2}] D^2 \phi [|D_i \phi|^{p/2}] = I_n$.

   A significant technical hurdle is that $\phi(x)$ is not $C^2$ on the hyperplanes $\{z_i = 0\}$, where the degeneracy occurs. To overcome this, the authors employ a dimensional reduction argument:

   • They decompose the set of touching points based on which coordinate directions are degenerate ($D_i u = 0$) and which are non-degenerate.
   • At points where specific derivatives vanish, they restrict the analysis to lower-dimensional slices (hyperplanes) where the remaining directions are non-degenerate.
   • On these slices, the restricted paraboloid is smooth, allowing the application of the area formula and standard measure estimates.
   • The results are then aggregated using Fubini's theorem.

   A similar strategy is applied to construct a barrier function $\Phi(x) = |\phi(x)|^{-a}$ to prove the doubling property, which is essential for the Harnack inequality.

Key Contributions and Results
The paper establishes the following fundamental results for viscosity solutions of the degenerate equations:

1. Aleksandrov-Bakelman-Pucci (ABP) Estimate (Theorem 1.1):
   The authors prove an ABP estimate bounding the supremum of the solution $u$ in terms of its boundary values and the $L^n$ norm of the source term $f$. Specifically:
   $$\sup_{\Omega} u \leq \sup_{\partial \Omega} u^+ + C \|f^-\|_{L^n(\Gamma^+(u))}^{\frac{1}{1+p}},$$
   where $\Gamma^+(u)$ is the upper contact set. This estimate is derived using the modified area formula involving the map $G$.

2. Harnack Inequality (Theorem 1.2):
   For non-negative solutions $u$ with $f \in L^n$, the paper proves a Harnack inequality:
   $$\sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + \|f\|_{L^n(B_1)}^{\frac{1}{1+p}} \right).$$
   This result is achieved by combining the measure estimate (Lemma 4.1) derived from the sliding anisotropic paraboloid with the doubling property (Lemma 5.1) derived from the anisotropic barrier.

3. Hölder Regularity (Corollary 1.1):
   As a direct consequence of the Harnack inequality, the authors establish the local Hölder continuity of solutions:
   $$\|u\|_{C^\alpha(B_{1/2})} \leq C \left( \|u\|_{L^\infty(B_1)} + \|f\|_{L^n(B_1)}^{\frac{1}{1+p}} \right).$$

Significance and Scope
The paper claims to provide the first basic regularity theory (ABP estimates and Harnack inequalities) for this specific class of degenerate fully nonlinear pseudo-$p$-Laplacian equations in nondivergence form. The authors note that while regularity for the divergence form is well-established (via Moser iteration), the nondivergence case presents unique challenges due to the lack of equivalence between the operator and the standard $p$-Laplacian in the nondivergence setting.

The significance lies in handling the unbounded coordinatewise degeneracy. The authors explicitly state that previous results for equations degenerating only where $|Du|$ is large do not apply here because the degeneracy set is much larger. The paper's contribution is the development of a tailored sliding paraboloid method using anisotropic functions and a dimensional reduction technique to manage the singularities at $D_i u = 0$.

The work is framed as a foundational step in the analysis of partial differential equations arising in anisotropic diffusion, orthotropic models, and transport phenomena in layered media, where the response depends on individual coordinate directions. The results are presented as establishing the "basic regularity" necessary for further analysis, without extending to higher-order regularity or specific physical applications beyond the mathematical framework established.