Harnack inequality for non-uniformly elliptic equations in non-divergence form

1. Problem Statement and Context

The paper investigates the regularity properties of solutions to degenerate elliptic equations in non-divergence form:
$$a_{ij}(x) \partial_{ij} u = 0 \quad \text{in } B_1 \subset \mathbb{R}^d$$
The coefficients satisfy the non-uniform ellipticity condition:
$$\lambda(x) I \leq a_{ij}(x) \leq I$$
where the ellipticity ratio $\lambda(x)$ is allowed to degenerate (approach zero). The central assumption is that the reciprocal of the ellipticity belongs to an $L^p$ space:
$$\lambda^{-1} \in L^p(B_1)$$
The paper seeks to determine for which values of $p$ classical regularity results (specifically Harnack inequalities, Weak Harnack inequalities, and Liouville theorems) hold. This contrasts with the uniformly elliptic case ($p=\infty$) and the divergence form case, where the critical threshold is known to be $p > \frac{d-1}{2}$.

Key Challenge: The degeneracy creates "traps" where diffusion is extremely slow, potentially disconnecting the values of the solution between different regions. The goal is to determine if the mixing effect remains strong enough to ensure comparability of solution values (Harnack inequality) despite these traps.

2. Methodology and Technical Approach

The author employs a combination of viscosity solution theory, geometric measure arguments, and barrier constructions.

• Viscosity Solutions: The results are established for viscosity solutions. To handle the lack of smoothness, the author utilizes infimal convolutions (for supersolutions) and supremal convolutions (for subsolutions) to approximate solutions by semi-concave or semi-convex functions, which are twice differentiable almost everywhere (Alexandroff's Theorem).
• Scale-Invariant Quantity: A key quantity introduced is $\Gamma(B)$, representing the average degeneracy on a ball $B$:
  $$\Gamma(B) := \left( \frac{1}{|B|} \int_B \lambda^{-p} \right)^{1/p}$$
  All estimates depend on this quantity, which can be large for small balls.
• Touching Arguments (Barrier Methods):
  • For Supersolutions: The author touches supersolutions from below using singular cusps of the form $|x|^{-\beta}$. Unlike previous works where $\beta$ was fixed, here $\beta$ scales with $\Gamma(B)$. This leads to a "doubling estimate" where the required lower bound on the solution grows exponentially with $\Gamma(B)$.
  • For Subsolutions: A novel approach involves touching subsolutions from above with double exponential blow-up barriers of the form $U(x) = c \exp(\exp((1-|x|^2)^{-1}))$. This allows the derivation of a logarithmic local maximum principle.
• Ink-Spot Arguments: The proof of the Weak Harnack inequality uses a modified "ink-spot" argument. Due to the degeneracy, the standard geometric iteration fails because the "bad" balls (high $\Gamma$) are simultaneously the most expensive in level (requiring large $M$) and the worst in measure gain. The author overcomes this by carefully tracking the dependence on $\Gamma(B)$.

3. Key Contributions and Results

A. Counterexamples and Optimality (Section 2)

The paper establishes sharp thresholds by constructing explicit counterexamples:

• No Power-Type $L^\epsilon$ Estimates: For any $p < \infty$, there is no estimate of the form $\int u^\epsilon \lesssim (\inf u)^\epsilon$. The best possible bound for supersolutions involves logarithmic moments.
• Failure for $p < d-1$: The author constructs solutions where the ratio $\sup u / \inf u \to \infty$ as the degeneracy concentrates, proving that neither Harnack nor Weak Harnack inequalities hold if $p < d-1$. This identifies $d-1$ as the critical threshold for the non-divergence case.

B. Log-$L^\epsilon$ Weak Harnack Inequality (Proposition 1.1)

For $p > d-1$, the paper proves a Weak Harnack inequality for non-negative supersolutions, but in a logarithmic form rather than a power form:
$$\int_{B_1} \left( \log \frac{u}{\inf_{B_1} u} \right)^\epsilon \leq C \Gamma(B_5)^\epsilon \gamma$$
where $\gamma = \frac{pd}{p-(d-1)}$. This is significantly weaker than the classical power-type Weak Harnack inequality but is the best possible given the counterexamples.

C. Improvement of Oscillation and Liouville Theorem (Proposition 1.2, Theorem 1.3)

Using the log-$L^\epsilon$ estimate, the author derives a unit-scale improvement of oscillation:
$$\text{osc}_{B_1} u \leq (1-\theta) \text{osc}_{B_5} u$$
where the decay factor $\theta$ depends exponentially on $\Gamma(B_5)$. This leads to a Liouville theorem for bounded solutions on $\mathbb{R}^d$ under the condition $p > d-1$.

D. Logarithmic Local Maximum Principle (Proposition 1.4)

A new result for subsolutions is derived by touching with double exponential barriers. It bounds the maximum of a subsolution in terms of its logarithmic moments:
$$\sup_{B_1} u \leq 2 \exp \left( c \Gamma(B_3)^{\dots} \left( \int_{\{u>1\}} (\log u)^\tau \right)^{\frac{1}{\tau - q\kappa}} \right)$$
This result holds for $p > d-1$ and is new even in the uniformly elliptic case.

E. Harnack Inequality (Theorem 1.5)

The main result establishes a Harnack inequality ($\sup_{B_1} u \leq C \inf_{B_1} u$) for non-negative solutions, but only under a stricter condition on $p$:
$$p > p(d) := \frac{d-1}{2} \left( d + 2 + \sqrt{d(d+4)} \right)$$
The constant $C$ depends exponentially on $\Gamma(B_5)$.

• Conjecture: The author conjectures that the Harnack inequality should hold for all $p > d-1$, matching the threshold for the Weak Harnack inequality. The gap between $d-1$ and $p(d)$ arises because the log-$L^\epsilon$ estimate (for supersolutions) and the logarithmic maximum principle (for subsolutions) are only compatible for sufficiently large $p$.

4. Significance and Implications

1. Sharp Thresholds: The paper rigorously identifies $p = d-1$ as the critical threshold for the existence of any form of Harnack or Weak Harnack inequality in the non-divergence setting. This contrasts with the divergence case ($p > \frac{d-1}{2}$), highlighting the fundamental difference in regularity theory between the two forms.
2. Logarithmic Estimates: The discovery that power-type $L^\epsilon$ estimates fail for any finite $p$ in the degenerate non-divergence case is a major theoretical insight. It suggests that solutions can be extremely "spiky," and only logarithmic control is possible.
3. New Barrier Techniques: The introduction of double exponential barriers for subsolutions provides a new tool for handling non-uniform ellipticity, potentially applicable to other degenerate problems.
4. Open Problems: The paper leaves open the range $d-1 < p \leq p(d)$, conjecturing that Harnack holds there. It also discusses the necessity of continuity and strict positivity of $\lambda$ for viscosity solutions, showing that for $L^s$-strong solutions ($s>d$), the strict positivity assumption can be dropped.

Summary Table of Results

Regime	Weak Harnack	Harnack	Notes
$p < d-1$	Fails	Fails	Counterexamples constructed; no integral estimates possible.
$d-1 < p \leq p(d)$	Holds (Log-type)	Conjectured	Log-$L^\epsilon$ and Local Max Principle hold; full Harnack unproven.
$p > p(d)$	Holds (Log-type)	Holds	Full Harnack inequality proven with exponential constant.
$p = \infty$	Holds (Power-type)	Holds	Classical uniformly elliptic case.

In conclusion, Bowman's work significantly advances the understanding of non-uniformly elliptic equations in non-divergence form, establishing that while regularity is preserved for $p > d-1$, it comes at the cost of weaker estimates (logarithmic vs. power) and requires a higher integrability threshold for the full Harnack inequality compared to the divergence case.