Harnack inequality for anisotropic fully nonlinear equations with nonstandard growth

1. Problem Statement

The paper addresses the regularity theory for a specific class of degenerate, fully nonlinear, anisotropic elliptic equations in non-divergence form. The primary equation under investigation is:

$$\begin{cases} M^-_{\lambda, \Lambda} \left( \left[ |D_i u|^{\frac{p_i-2}{2}} \right] D^2 u \left[ |D_i u|^{\frac{p_i-2}{2}} \right] \right) - \mu \sum_{i} |D_i u|^{p_i-1} \leq c_0, \\ M^+_{\lambda, \Lambda} \left( \left[ |D_i u|^{\frac{p_i-2}{2}} \right] D^2 u \left[ |D_i u|^{\frac{p_i-2}{2}} \right] \right) + \mu \sum_{i} |D_i u|^{p_i-1} \geq -c_0, \end{cases}$$

where:

• $M^\pm_{\lambda, \Lambda}$ are the Pucci extremal operators.
• $p_i \geq 2$ are distinct exponents for each coordinate direction, leading to anisotropic degeneracy (degeneracy occurs if any single partial derivative vanishes, unlike the standard $p$-Laplacian where the whole gradient must vanish).
• The equation exhibits non-standard growth, meaning the growth rate varies between $p_1$ and $p_n$ (where $2 \leq p_1 \leq \dots \leq p_n$).
• The solutions considered are viscosity solutions ($u \in C(\Omega)$), not weak solutions.

The Core Challenge:
Establishing the Harnack inequality (and consequently Hölder continuity) for viscosity solutions of this system. Previous methods for weak solutions (De Giorgi-Nash-Moser theory) rely on Sobolev inequalities and variational structures, which are unavailable for viscosity solutions in this non-divergence, anisotropic setting. Furthermore, the anisotropic scaling makes standard covering arguments and barrier constructions extremely difficult.

2. Methodology

The authors employ a combination of intrinsic geometry techniques and viscosity solution methods, adapted specifically for the anisotropic nature of the operator.

A. Intrinsic Scaling and Geometry

Due to the non-homogeneous scaling of the equation, the authors utilize intrinsic cubes $K_r(M)$ rather than standard Euclidean cubes. The intrinsic cube is defined as:
$$K_r(M) = \{ x \in \mathbb{R}^n : |x_i| \leq r^{\alpha_i} M^{\beta_i} \}$$
where $\alpha_i = 1/p_i$ and $\beta_i = -(p_n - p_i)/p_i$. This scaling matches the natural behavior of the equation at different height levels $M$.

B. Sliding Paraboloid Method (Measure Estimates)

To derive basic measure estimates (Lemma 3.1), the authors adapt the sliding paraboloid method (originally by Cabré and Savin).

• Adaptation: Instead of a standard isotropic paraboloid, they use an anisotropic paraboloid of the form $\phi(x) = \sum K_i |x_i|^{p_i/(p_i-1)}$.
• Technical Handling: Since this function is not $C^2$ on coordinate hyperplanes, they utilize a "restriction to slices" argument to handle the singularities, ensuring the area formula can be applied to estimate the measure of contact points.

C. Construction of a Novel Barrier Function

A central contribution of the paper is the explicit construction of a barrier function to prove the doubling property (Lemma 5.1).

• Form: The barrier is defined as $\Phi(x) = (|b_1 x_1|^{a_1} + \dots + |b_n x_n|^{a_n})^{-\gamma}$.
• Role: This function serves as a viscosity subsolution to the equation. By carefully selecting parameters $a_i$ and $\gamma$ based on the exponents $p_i$, the authors ensure the barrier satisfies the necessary differential inequalities.
• Significance: This explicit construction overcomes the lack of an abstract functional-analytic framework (like the Barenblatt solution used in some weak solution contexts) and allows for the comparison principle to be applied effectively.

D. Iteration and Covering Arguments

The proof combines the measure estimates and the doubling property using Vitali covering lemmas adapted to intrinsic cubes. This leads to $L^\epsilon$ estimates (Theorem 6.1), which quantify the algebraic decay of the level sets $\{u > M\}$.

3. Key Contributions

1. First Harnack Inequality for Viscosity Solutions: This is the first result establishing the intrinsic Harnack inequality for viscosity solutions of degenerate anisotropic fully nonlinear equations with non-standard growth.
2. Novel Barrier Construction: The explicit construction of the anisotropic barrier function $\Phi(x)$ is a significant technical innovation. It allows the derivation of the doubling property without relying on the existence of weak solutions or Sobolev embeddings.
3. Adaptation of Sliding Paraboloid: Successfully adapting the sliding paraboloid method to handle the strong anisotropic degeneracy and the lack of $C^2$ regularity in the test functions.
4. Intrinsic Geometry Framework: Rigorous application of intrinsic scaling to handle the distinct growth rates in every coordinate direction, a situation more complex than the parabolic $p$-Laplacian (which only has time vs. space scaling).

4. Main Results

Theorem 1.1 (Intrinsic Harnack Inequality)

Under specific conditions on the exponents $p_i$, if $u \geq 0$ is a viscosity solution, then there exist constants $C_0, R_0 > 1$ such that:

1. Upper Bound: $\sup_{K_r(u(0))} u \leq C_0 u(0)$.
2. Lower Bound: $\inf_{K_r(C_0 u(0))} u \geq \frac{1}{C_0} u(0)$.

Condition on Exponents:
The result holds if the gap between the smallest and largest exponents satisfies:
$$\frac{p_n - 1}{p_1 - 1} \leq p_n \frac{n - \lambda/\Lambda}{n}$$
where $p = (\frac{1}{n} \sum \frac{1}{p_i})^{-1}$ is the harmonic mean. This condition ensures the "gap" between $p_1$ and $p_n$ is not too large, preventing counterexamples to regularity.

Corollary 1.1 (Hölder Regularity)

As a direct consequence, viscosity solutions are locally Hölder continuous with an exponent $\alpha \in (0, 1)$ depending on $n, (p_i), \lambda, \Lambda$.

5. Significance and Context

• Bridging a Gap: Prior to this work, Harnack inequalities for this specific class of equations were known for weak solutions (under certain conditions) but remained an open problem for viscosity solutions due to the lack of variational structure.
• Handling Anisotropy: The paper demonstrates that even with strong anisotropic degeneracy (where degeneracy can occur in just one direction), regularity can be preserved provided the exponents are sufficiently close.
• Counterexamples: The authors note that if the gap between $p_1$ and $p_n$ is too large, counterexamples exist (unbounded solutions), highlighting the optimality of their structural condition.
• Future Applications: The explicit barrier construction developed here is anticipated to be useful for establishing Harnack inequalities for weak solutions in similar anisotropic settings, potentially relaxing existing assumptions.

In summary, this paper provides a rigorous regularity theory for a challenging class of nonlinear PDEs by synthesizing intrinsic geometry, viscosity solution techniques, and novel barrier constructions, thereby extending the scope of Harnack inequalities to fully nonlinear anisotropic operators.