Harnack Inequality for Weak Solutions

• Harnack inequality is a quantitative estimate that controls the oscillation of nonnegative weak solutions to elliptic and parabolic PDEs.
• It links the infimum over larger domains with the supremum on smaller ones, playing a critical role in regularity theory under minimal integrability and structural hypotheses.
• The methodology employs energy estimates, reverse Hölder inequalities, and intrinsic scaling, extending to degenerate, nonlocal, and variable-exponent settings.

A Harnack inequality provides quantitative control over the oscillation of nonnegative weak solutions to elliptic or parabolic partial differential equations (PDEs), linking infima and averages or suprema in separated sets. For weak solutions—interpreted in a variational or viscosity sense rather than pointwise—the inequality is central to the regularity theory of degenerate, nonuniformly elliptic, singular, and/or nonlocal PDEs, under minimal structure and integrability hypotheses.

1. Classical Setting and Foundational Results

The classical Harnack inequality asserts that if $u$ is a nonnegative weak solution to a uniformly elliptic or parabolic equation in divergence form, then in any pair of nested domains (cylinders or balls), the supremum of $u$ on the smaller is bounded by a constant times the infimum on the larger. The quantitative constant depends on dimension, ellipticity parameters, and geometric ratios, but not on $u$ itself—this is due to the intrinsic scaling of the PDE.

For example, for weak solutions $u\ge0$ to $\operatorname{div}(A(x)\nabla u)=0$ in a ball $B_{2R}$ with $A$ uniformly elliptic, Moser's theory and its variants yield

$\sup_{B_R} u \leq C\,\inf_{B_R}u,$

where $C$ depends only on $n$ and the ellipticity constants. In the parabolic case, analogous estimates relate suprema and infima across intrinsic space-time cylinders shifted in time.

The extension of this theory to settings with nonuniform ellipticity, time dependence, degeneracies, nonlocalities, or superlinear lower-order terms requires significant adaptations to the variational or weak solution framework.

2. Weak Harnack Inequality: Definitions and General Form

Let $u$ be a (nonnegative) weak supersolution to a PDE in a domain $Q$. A weak Harnack inequality is a reverse Hölder or geometric mean control of $u$ in terms of its pointwise infimum: $\Bigl( \fint_{Q_1} u^\varepsilon\,dz \Bigr)^{1/\varepsilon} \leq C \Bigl( \inf_{Q_2} u + \text{lower order terms} \Bigr),$ where $Q_1\subset Q_2\subset Q$ are nested cylinders (balls, space-time rectangles, etc.), $C > 0$, $\varepsilon\in(0,1)$, and the 'lower order' may include tail or $L^p$-norm terms if $f$ or nonlocalities are present.

This estimate is valid in a wide range of PDE settings:

• Uniformly elliptic equations with unbounded lower-order coefficients and source: Proven for $L^p$-viscosity supersolutions to fully nonlinear parabolic PDEs with $f\in L^p$, $\mu\in L^q$ under $q> n+2$, $p>p_1(n,\lambda,\Lambda)$

$u_t + P^+(D^2u) + \mu(x,t)|Du| + f(x,t) = 0$

The central result establishes (for small $\|f\|_{L^p}$)

$\left(\fint_{J_1} u^{\epsilon_0}\right)^{1/\epsilon_0} \leq C_0 \left( \inf_{J_2} u + \|f\|_{L^p(Q)} \right)$

with explicit constants and exponents depending on the dimension, ellipticity, $L^p$-integrability, and lower-order norm bounds (Koike et al., 2018).

• Degenerate equations—quasilinear, weighted, or with rough coefficients: For divergence-form operators with measurable coefficients or degenerate quadratic forms, under a local homogeneous Sobolev/Poincaré structure, weak solutions admit analogous estimates:

$\sup_{B_R}(u+k) \leq C \inf_{B_R}(u+k)$

with $k$ shifts absorbing lower-order terms (1411.69761010.0322).

• Generalized growth—Orlicz, Musielak-Orlicz, or variable-exponent structure: For divergence-form equations

$-\operatorname{div}f(x,\nabla u) = 0$

with $p$- and $q$-type Orlicz growth and coercivity, the weak Harnack is formulated in terms of Orlicz means and modular functionals, with the optimal range determined by the growth exponents and local regularity (Benyaiche et al., 2020Shan et al., 2020).

• Nonlocal and Fractional Operators: In both elliptic and parabolic nonlocal equations (fractional Laplacians, mixed local/nonlocal, variable-exponent or double-phase growth), weak Harnack inequalities feature a 'tail' term reflecting the interaction of $u$ outside the reference ball:

$\left( \fint_{B_r} u^{-\varepsilon}\right)^{-1/\varepsilon} \leq C \left(\inf_{B_{r/2}} u + \text{tail term}\right)$

The tail may involve e.g.\ $$r^{sp} \int_{\mathbb{R}^n\setminus B_r} g(|u(y)|)/|y-x_0|^{n+sp} \, dy$$ for the $G$-Laplacian (Fang et al., 2022Garain, 5 Oct 2025Chaker et al., 2022).

• Degenerate Kolmogorov (ultraparabolic) Equations: Using specialized function spaces $\mathcal{W}$ adapted to the possibly degenerate operator, and novel group-theoretic techniques, one obtains

$\left(\fint_{Q_-} u^p\right)^{1/p} \leq C\,\inf_{Q_+} u + C \|f\|_{L^q}$

for any $p > 1/2$ as determined by the integrability of the lower-order terms (Anceschi et al., 2021).

3. Methodologies: Iterative, Functional, and Geometric Techniques

The proof architecture for weak Harnack inequalities is uniform across settings, but with essential adaptations:

1. Energy (Caccioppoli) Estimates: Testing the equation against properly shifted/truncated or powered test functions (typically $(u + \varepsilon)^{-k}\psi^p$) yields integrability and mean-control of $u$ or $\log u$.
2. Reverse Hölder and John–Nirenberg/Bombieri–Giusti Lemmas: The reverse Hölder inequalities control high integrability from low (or vice versa), while the exponential integrability of $\log u$ (John–Nirenberg) is exploited to transfer $L^p$– to $L^-p$–type controls and handle non-uniform settings. These tools are crucial for both local and nonlocal (fractional or Orlicz) PDEs (Garain, 5 Oct 2025Benyaiche et al., 2020).
3. Expansion of Positivity and Measure-to-Point Estimates: De Giorgi-type or Krylov–Safonov covering arguments allow translation from a high-density positivity set to quantitative lower bounds, essential in both local and nonlocal theories.
4. Intrinsic Scaling: For degenerate equations (e.g., $p$-parabolic), cylinders are scaled according to the solution's size: intrinsic time scales such as $\theta\,r^p$ with $\theta\sim u(x_0,t_0)^{-(p-2)}$ are crucial for capturing correct regularity (Arya et al., 12 Jun 2025).
5. Nonlocal Tail Terms: For nonlocal equations, energy estimates and weak Harnack must include explicit control on the tail, encoding long-range interactions.
6. Weak Poincaré and Sobolev Embeddings: In ultraparabolic or rough coefficient settings, standard Sobolev inequalities fail; adapted forms in homogenized or dual norms, often via group structure, are used (Anceschi et al., 2021).

4. Extensions: Inhomogeneities, Nonstandard Growth, and Nonlocality

Recent advances establish weak Harnack inequalities in settings previously inaccessible to classical methods:

• Fully Nonlinear (Nondivergence) Degenerate Parabolic Equations: Arya–Julin establishes a sharp weak Harnack for viscosity supersolutions to

$u_t - F(D^2u,Du,x,t) = 0,$

in domains and cylinders adapted to the intrinsic scaling, with exponents and constants explicit in the ellipticity and nonlinearity parameters (Arya et al., 12 Jun 2025).

• Equations with Orlicz or Variable-Exponent Structure: Chaker–Kim–Weidner prove weak and full Harnack for minimizers and weak solutions of nonlocal equations with general homogeneous and inhomogeneous Orlicz-type growth, including explicit tails and sharp exponent restrictions (Chaker et al., 2022, Benyaiche et al., 2020, Shan et al., 2020).
• Mixed Local/Nonlocal and Nonhomogeneous Problems: Garain demonstrates two new analytic methods (John–Nirenberg and Bombieri–Giusti) to obtain weak Harnack inequalities, even for mixed $p$-Laplace and fractional $p$-Laplace equations with nonzero source, bypassing traditional positivity expansion and covering lemmas (Garain, 5 Oct 2025).
• Kolmogorov and Kinetic Equations: Anceschi–Rebucci adapt the Kruzkov log-transform method and introduce the 'Ink-Spots' covering theorem for slanted cylinders in the study of degenerate Kolmogorov equations with inhomogeneous and drift terms (Anceschi et al., 2021).
• Nonlocal Parabolic and Fractional-Time Equations: Strömqvist, Kassmann–Weidner, and others provide precise weak Harnack (and full Harnack) inequalities for weak solutions to parabolic equations with nonlocal operators, including sharp treatment of the 'tail' and precise dependence on nonlocal interaction/ellipticity (Kassmann et al., 2023Strömqvist, 2018Zacher, 2010).

5. Consequences, Sharpness, and Applications

Table: Paradigmatic Weak Harnack Inequalities

Class of PDE	Weak Harnack Prototype	Reference
Fully nonlinear parabolic, $L^p$-viscosity	$\left( \fint_J u^{\epsilon_0}\right)^{1/\epsilon_0} \leq C\left(\inf_{J'} u + \|f\|_{L^p}\right)$	(Koike et al., 2018)
Degenerate quasilinear (metric/Q weight)	$\sup_{B_R} (u+k) \leq C \inf_{B_R} (u+k)$	(Monticelli et al., 2014)
Nonlocal, $p$-fractional growth	$\left( \fint_{B_r} u^{-\varepsilon} \right)^{-1/\varepsilon} \leq C ( \inf_{B_{r/2}} u + \text{tail} )$	(Fang et al., 2022, Garain, 5 Oct 2025)
Kolmogorov/ultraparabolic	$\left( \fint_{Q_-} u^p \right)^{1/p} \leq C ( \inf_{Q_+} u + \|f\|_{L^q} )$	(Anceschi et al., 2021)
Doubly nonlinear, mixed local/nonlocal	$u\geq \gamma^{-1}u(x_0,t_0)$ in $Q^+$, in intrinsic cylinders	(Radulescu et al., 2024)

Consequences of weak Harnack inequalities include:

• Full Harnack inequalities: By coupling weak Harnack with local sup (maximum) principles/subsolution bounds, one obtains two-sided oscillation control:

$$\sup_{J_3}u \leq C\left(\inf_{J_2} u + \|f\|_{L^p}\right)$$

for solutions, with explicit constants/hypotheses (Koike et al., 2018Arya et al., 12 Jun 2025).

• Hölder continuity: Via oscillation-decay (Campanato iteration), weak Harnack directly implies local Hölder regularity for bounded nonnegative weak solutions, even with degeneracy, nonlocality, or nonuniform structural data (Koike et al., 20181411.69762510.04065).
• Strong maximum principles and Liouville theorems: The propagation of positivity property can be used, via weak Harnack, to establish impossibility of achieving isolated nontrivial maxima or minima, leading to classification of entire solutions (Goering, 20201009.48521503.04472).
• Sharpness and optimality: Counterexamples (e.g., double-phase with critical exponents, or variable exponent with insufficient regularity) demonstrate necessity of structural or integrability assumptions. The optimal exponents for $L^p$ and Orlicz growth are tracked and known to be best possible (Benyaiche et al., 2020).

6. Structural and Geometric Hypotheses

The minimum requirements for weak Harnack are determined by the analytic and geometric features of the operator and space:

• Ellipticity/degeneracy: Either uniform (Pucci operators), degenerate quadratic forms, or anisotropic (Finsler norms) are allowed, provided measurable control and comparability (Koike et al., 2018Goering, 2020Monticelli et al., 2014).
• Volume doubling and Poincaré/Sobolev inequalities: In metric spaces or rough-geometry settings (e.g., subelliptic or fractal), weak Harnack is valid when these properties are present (1205.64931708.06329).
• Nonlocal kernel control: For fractional or nonlocal problems, power-type bounds and symmetry for the kernel, along with growth conditions on the nonlinearity or Orlicz function, are essential for tail estimates and Moser iteration (Fang et al., 2022Kassmann et al., 2023).
• Lower-order integrability: For terms depending on $\mu(x,t)$, $f$, etc., integrability conditions relating to dimensionality are sharp and necessary for both energy estimates and maximal inequalities (Koike et al., 2018Anceschi et al., 2021).

7. Connections and Future Directions

The weak Harnack inequality for weak solutions has become a central pillar in the theory of nonlinear and nonlocal regularity. Its reach spans local and nonlocal, degenerate, variable-exponent, rough-geometry, kinetic, and stochastic PDEs. Key future interests include:

• Extending to nonconvex, rapidly oscillatory, or measure-valued coefficients;
• Tighter quantitative dependence of constants on inhomogeneities, dimension, and operator structure;
• Nonstandard growth (double-phase, Orlicz, nonseparable functionals) in higher-order and vectorial systems;
• Deepening understanding of stochastic, kinetic, and ultraparabolic models via weak Harnack and associated probabilistic frameworks.

The weak Harnack inequality both reflects and enables the propagation of positivity and oscillation control vital to qualitative and quantitative analysis of weakly-defined PDE solutions in the broadest contemporary sense.

Key references: (Koike et al., 2018, Benyaiche et al., 2020, Monticelli et al., 2014, Garain, 5 Oct 2025, Radulescu et al., 2024, Anceschi et al., 2021, Kassmann et al., 2023, Fang et al., 2022, Fazio et al., 2010, Zacher, 2010, Goering, 2020).