Harnack's inequality

Harnack's inequality is a classical result in mathematics that bounds the ratio of the maximum and minimum values of a non-negative harmonic function over connected subsets of its domain, ensuring that such functions cannot vary too wildly without changing sign.^[1] Introduced by the German mathematician Carl Gustav Axel von Harnack in 1887, the inequality originally applied to positive harmonic functions in the plane, derived using the Poisson integral formula.^[1] In its general form for dimensions d \geq 2, for a non-negative harmonic function u on a ball B_R(x_0) \subset \mathbb{R}^d, the inequality states that for any x \in B_r(x_0) with r < R,

u(x_0) \left( \frac{R - r}{R + r} \right)^{d-1} \leq u(x) \leq u(x_0) \left( \frac{R + r}{R - r} \right)^{d-1},

providing explicit control on how much u can oscillate within smaller balls centered at x_0.^[1] This result implies the Harnack maximum principle, which states that the maximum of a non-negative harmonic function on a domain is attained on the boundary, and has profound implications for the regularity and behavior of solutions to elliptic partial differential equations (PDEs).^[2] Over the decades, Harnack's inequality has been extended far beyond classical harmonic functions to encompass solutions of more general elliptic and parabolic PDEs.^[2] Jürgen Moser's 1961 work generalized it to non-negative solutions of uniformly elliptic equations in divergence form, establishing a constant C > 0 such that \sup_{B_{r/2}} u \leq C \inf_{B_{r/2}} u for balls B_r in the domain.^[2] Further advancements by N. V. Krylov and M. V. Safonov in 1980 extended the inequality to non-divergence form operators, while J. Serrin in 1964 and N. S. Trudinger in 1967 adapted it to quasilinear elliptic equations, possibly degenerate.^[2] These extensions have applications in proving Liouville theorems (bounding entire solutions), strong maximum principles, and gradient estimates, making Harnack inequalities indispensable tools in analysis on Riemannian manifolds, nonlocal operators like the fractional Laplacian, and even discrete settings such as graphs.^[1]

Introduction and History

Overview and Motivation

Harmonic functions are real-valued, twice continuously differentiable functions u defined on an open domain \Omega \subset \mathbb{R}^n that satisfy Laplace's equation \Delta u = 0, where \Delta denotes the Laplacian operator.^[3] These functions model steady-state solutions in physical contexts, such as electrostatic potentials or temperature distributions in the absence of sources.^[4] Harnack's inequality applies to non-negative harmonic functions in connected domains, providing pointwise upper and lower bounds on their values at interior points relative to each other.^[3] This ensures that such functions cannot exhibit wild oscillations within compact subdomains, maintaining a controlled variation that reflects their analytic smoothness.^[4] Geometrically, the inequality conveys that harmonic solutions evolve gradually, with values in smaller subdomains bounded by factors depending on the domain's scale, preventing rapid changes over short distances.^[3] In potential theory, Harnack's inequality motivates the study of how interior values relate to boundary data, allowing estimates of maxima and minima without solving the boundary value problem explicitly.^[3] For instance, it implies that if a non-negative harmonic function attains zero at one interior point, it must be identically zero nearby, highlighting the rigidity of these solutions.^[4] A simple illustration occurs in one dimension, where solutions to u'' = 0 are linear functions on an interval. For non-negative u on (-r, r), the inequality yields bounds like u(x) \leq \frac{r + d}{r - d} u(y) for y at the center and |x - y| = d < r, demonstrating controlled growth.^[3] This principle extends briefly to generalizations for elliptic and parabolic partial differential equations, offering analogous bounds on positive solutions.^[4]

Historical Development

Harnack's inequality traces its roots to foundational studies in potential theory during the early 19th century, where the mean value property of harmonic functions played a central role. Carl Friedrich Gauss and Siméon Denis Poisson established key properties of harmonic functions, including the mean value theorem, which asserts that the value of a harmonic function at a point equals its average over any sphere centered at that point. These insights, developed in the context of gravitational and electrostatic potentials, provided the groundwork for later bounds on harmonic functions, as harmonic functions satisfy Laplace's equation and exhibit maximum principles that limit oscillations.^[3]^[5] The inequality itself was formally introduced by Axel Harnack in 1887, in his seminal book Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, where he derived bounds relating the values of positive harmonic functions at points within a domain, building directly on the mean value property to quantify how much such functions can vary. Harnack's work focused on two-dimensional potential theory and integral representations, establishing that for a positive harmonic function in a ball, the ratio of its maximum to minimum values is controlled by the domain's geometry. This result marked a milestone in elliptic partial differential equations, providing a tool for regularity estimates.^[6] In the mid-20th century, the inequality saw significant generalizations to broader classes of elliptic equations. James Serrin extended Harnack's result in 1955 to positive solutions of uniformly elliptic linear equations with continuous coefficients, relaxing assumptions on the domain and operator while preserving the core bounding mechanism. This paved the way for Jürgen Moser's influential contributions in 1960 and 1964, where he established Harnack inequalities for weak solutions to elliptic equations in divergence form, leveraging the De Giorgi-Nash theory to prove higher regularity without classical smoothness assumptions on solutions. Moser's approach, using iteration techniques on test functions, demonstrated local boundedness and Hölder continuity for weak solutions, fundamentally advancing the regularity theory for nonlinear elliptic problems. The evolution continued into parabolic settings during the mid-20th century, extending Harnack's ideas to time-dependent equations like the heat equation. Building on studies of diffusion processes and the fundamental solution of the heat equation, researchers adapted the inequality to nonnegative solutions of parabolic equations, yielding estimates that control spatial and temporal variations simultaneously. Moser's 1964 work on parabolic Harnack inequalities further solidified this, applying to weak solutions of the heat equation and related operators, which proved essential for analyzing evolution equations.^[7] In modern developments, Harnack inequalities found profound applications in geometric analysis. Richard Hamilton in 1993 derived a matrix Harnack estimate for the Ricci flow, a nonlinear parabolic equation deforming Riemannian metrics, which bounds the evolution of curvature tensors and ensures short-time existence of smooth flows. This estimate was instrumental in Grigori Perelman's 2002–2003 proof of the Poincaré conjecture, where Perelman employed entropy functionals and monotonicity formulas inspired by Hamilton's Harnack inequality to control singularities in three-dimensional Ricci flow, ultimately verifying the geometrization of three-manifolds.

Core Results for Harmonic Functions

Statement of the Inequality

Harnack's inequality quantifies the controlled oscillation of non-negative harmonic functions within suitable subdomains of Euclidean space. Specifically, let u be a non-negative harmonic function on the open ball B_R(x_0) \subset \mathbb{R}^n for R > 0. Then, for every x \in B_r(x_0) where $0 < r < R,

\left( \frac{R - r}{R + r} \right)^{n-1} u(x_0) \leq u(x) \leq \left( \frac{R + r}{R - r} \right)^{n-1} u(x_0).

The constant in this bound arises from applying the mean value property to radially symmetric harmonic functions, which achieve the extremal ratios through explicit solutions involving powers of the radius.^[3] In the special case n=1, where harmonic functions are linear, the exponent vanishes, yielding the explicit form \frac{R - |x - x_0|}{R + |x - x_0|} u(x_0) \leq u(x) \leq \frac{R + |x - x_0|}{R - |x - x_0|} u(x_0). A direct consequence for concentric subballs is that \sup_{B_{R/2}(x_0)} u \leq 3^{n} \inf_{B_{R/2}(x_0)} u, as the maximum ratio over points in B_{R/2}(x_0) relative to the center (or infimum) is attained near the boundary of this smaller ball.^[3] For a more general setting, consider a connected open domain \Omega \subset \mathbb{R}^n. Harnack's inequality asserts that for any compact set K \subset \Omega, there exists a constant C_K > 0 (depending only on K and \Omega) such that \sup_K u \leq C_K \inf_K u for every non-negative harmonic function u on \Omega. When \Omega is bounded and K is covered by balls of radius \rho at distance d > \rho from \partial \Omega, the constant can be taken as C_K = \left( \frac{d + \rho}{d - \rho} \right)^{n-1}.^[8]^[3] This local boundedness control implies Harnack's theorem: any sequence of non-negative harmonic functions on \Omega that is locally bounded converges uniformly on compact subsets of \Omega to a non-negative harmonic function.^[3]

Proof in a Ball

Consider a non-negative function u that is harmonic in the ball B_r(0) \subset \mathbb{R}^n for n \geq 2. Assume u extends continuously to the closed ball \overline{B_r(0)}. The Poisson integral formula represents u inside the ball as

u(x) = \int_{\partial B_r(0)} P(x, y) \, u(y) \, d\sigma(y),

where d\sigma is the surface measure on the sphere \partial B_r(0), and the Poisson kernel is

P(x, y) = \frac{r^2 - |x|^2}{\omega_n r \, |x - y|^n}

for n \geq 3, with \omega_n denoting the surface area of the unit sphere in \mathbb{R}^n. For n = 2, the kernel takes the explicit form

P(x, y) = \frac{r^2 - |x|^2}{2\pi r \, |x - y|^2}.

This kernel satisfies \int_{\partial B_r(0)} P(x, y) \, d\sigma(y) = 1 for all x \in B_r(0), ensuring the representation holds.^[3] To derive Harnack's inequality in the inner ball B_{r/2}(0), first evaluate at the center: u(0) = \frac{1}{\omega_n r^{n-1}} \int_{\partial B_r(0)} u(y) \, d\sigma(y), since P(0, y) is constant and equal to $1/(\omega_n r^{n-1}). For general x \in B_r(0),

\frac{P(x, y)}{P(0, y)} = (r^2 - |x|^2) r^{n-2} / |x - y|^n.

For |x| \leq \rho < r and |y| = r, the distance satisfies r - |x| \leq |x - y| \leq r + |x|, so (r - |x|)^n \leq |x - y|^n \leq (r + |x|)^n. Thus,

\frac{r^2 - |x|^2}{ (r + |x|)^n } r^{n-2} \leq \frac{P(x, y)}{P(0, y)} \leq \frac{r^2 - |x|^2}{ (r - |x|)^n } r^{n-2}.

Factoring r^2 - |x|^2 = (r - |x|)(r + |x|) yields

\frac{r - |x|}{r + |x|} \left( \frac{r}{r + |x|} \right)^{n-2} \leq \frac{P(x, y)}{P(0, y)} \leq \frac{r + |x|}{r - |x|} \left( \frac{r}{r - |x|} \right)^{n-2}.

Let t = |x|/r \leq 1/2 for x \in B_{r/2}(0). The lower bound simplifies to (1 - t)/(1 + t)^{n-1} and the upper to (1 + t)/(1 - t)^{n-1}.^[3] Now, u(x) = \int_{\partial B_r(0)} \frac{P(x, y)}{P(0, y)} \, [P(0, y) u(y)] \, d\sigma(y). The measure \mu(dy) = P(0, y) \, d\sigma(y) is a probability measure on \partial B_r(0), so u(0) = \int u(y) \, \mu(dy). Let k(x, y) = P(x, y)/P(0, y), bounded between \alpha(t) = (1 - t)/(1 + t)^{n-1} and \beta(t) = (1 + t)/(1 - t)^{n-1}. Then,

\alpha(t) \, u(0) \leq u(x) \leq \beta(t) \, u(0).

For t \leq 1/2, the minimal \alpha(1/2) = (1/2)/(3/2)^{n-1} = 2^{n-2}/3^{n-1} and maximal \beta(1/2) = (3/2)/(1/2)^{n-1} = 3^{n-1}/2^{n-2} hold uniformly over B_{r/2}(0). Thus, \inf_{B_{r/2}(0)} u \geq \alpha(1/2) u(0) and \sup_{B_{r/2}(0)} u \leq \beta(1/2) u(0). Since u(0) \leq \sup_{B_{r/2}(0)} u / \beta(1/2) wait, no: from the lower bound, u(0) \leq \inf_{B_{r/2}(0)} u / \alpha(1/2), so \sup_{B_{r/2}(0)} u \leq \beta(1/2) u(0) \leq \beta(1/2) \cdot (\inf_{B_{r/2}(0)} u / \alpha(1/2)) = [\beta(1/2)/\alpha(1/2)] \inf_{B_{r/2}(0)} u. The ratio satisfies

\frac{\sup_{B_{r/2}(0)} u}{\inf_{B_{r/2}(0)} u} \leq \frac{\beta(1/2)}{\alpha(1/2)} = \left( \frac{1 + 1/2}{1 - 1/2} \right)^n = 3^n.

This establishes Harnack's inequality in the ball with the explicit constant $3^n.^[3]

Generalizations to Partial Differential Equations

Elliptic Equations

Harnack's inequality extends naturally to positive solutions of linear elliptic partial differential equations, serving as a cornerstone for regularity theory in this setting. Consider the operator L u = \sum_{i,j=1}^n a_{ij}(x) \partial_{ij} u + \sum_{i=1}^n b_i(x) \partial_i u + c(x) u, where the coefficients a_{ij}, b_i, and c are bounded, and the principal part satisfies the uniform ellipticity condition \lambda |\xi|^2 \leq \sum_{i,j=1}^n a_{ij}(x) \xi_i \xi_j \leq \Lambda |\xi|^2 for all x in the domain and all \xi \in \mathbb{R}^n, with $0 < \lambda \leq \Lambda < \infty. This framework encompasses the Laplacian as a special case when the coefficients simplify accordingly.^[9] For a nonnegative weak solution u to Lu = 0 in a ball B_r(x_0) \subset \mathbb{R}^n, the Harnack inequality asserts that

\sup_{B_{r/2}(x_0)} u \leq C \inf_{B_{r/2}(x_0)} u,

where the constant C depends only on the dimension n, the ellipticity ratio \lambda / \Lambda, the sup-norms \|b\|_\infty and \|c\|_\infty, and the radius r.^[9] This bound controls the oscillation of u in smaller concentric balls, providing essential interior regularity information.^[10] The result traces back to Serrin's 1955 work, which established the inequality for strong solutions under suitable assumptions on the coefficients.^[10] In 1961, Moser extended it to weak solutions, leveraging the De Giorgi-Nash framework for Hölder continuity of solutions to uniformly elliptic equations with measurable coefficients.^[9] De Giorgi's 1957 analysis for divergence-form equations and Nash's 1958 contributions for non-divergence form laid the groundwork by proving higher integrability and continuity, enabling Moser's Harnack refinement. A key proof technique is Moser's iteration method, which constructs a sequence of L^p-estimates for subsolutions starting from an initial L^2-bound (often from energy estimates or maximum principles) and iterating to L^\infty-control.^[9] This involves multiplying the weak form of the equation by test functions like u^{p-1} \eta^2 (with \eta a cutoff), integrating by parts, applying the ellipticity to absorb terms, and invoking Sobolev inequalities to bootstrap the exponents p. A related iteration yields the reverse Harnack for supersolutions, combining to the full inequality.^[9] These interior estimates hold independently of boundary data, relying solely on the local behavior of the solution and coefficients within the ball, which underscores their utility in proving global regularity from local control.^[10]

Parabolic Equations

Harnack's inequality adapts to non-negative solutions of linear parabolic partial differential equations defined in space-time domains, such as the cylinder Q_T = B_r \times (0, T), where B_r denotes the ball of radius r in \mathbb{R}^n. Consider the equation

\partial_t u - \operatorname{div}(A \nabla u) + b \cdot \nabla u + c u = 0

in Q_T, where A = (a_{ij}) is a symmetric matrix-valued function that is uniformly elliptic, meaning there exist positive constants \lambda, \Lambda such that \lambda |\xi|^2 \leq a_{ij} \xi_i \xi_j \leq \Lambda |\xi|^2 for all \xi \in \mathbb{R}^n, and the coefficients b, c are bounded.^[11] The forward Harnack inequality provides a bound on the value of such a solution at an earlier time by its value at a later time within a smaller cylinder. Specifically, for non-negative solutions u, there exists a constant C > 0 depending only on the dimension n, the ellipticity constants \lambda, \Lambda, the bounds on \|b\|_\infty and \|c\|_\infty, and the ratio T/r^2, such that for all (x, t), (y, s) \in Q_{T/2} with s \geq t,

u(x, t) \leq C \, u(y, s).

This result was established by Moser using an iteration technique based on L^p-estimates and the John-Nirenberg lemma adapted to the parabolic setting.^[11] A backward Harnack inequality holds for ancient solutions, which are non-negative solutions defined on unbounded backward time intervals, such as Q_\infty = B_r \times (-\infty, 0]. In this case, the inequality reverses the time direction, bounding the solution at later times by its values at sufficiently early times, with the constant C similarly depending on the structural parameters of the equation and the spatial scale r. This form is crucial for analyzing long-time behavior and was developed by Fabes, Garofalo, and Salsa through potential-theoretic methods and boundary behavior controls.^[12] The proofs of these inequalities rely on the strong maximum principle for parabolic equations, which ensures that non-negative solutions achieve their maxima on the parabolic boundary, combined with scaling arguments that homogenize the domain to unit size. The explicit dependence of the constant C on the parameters arises from Gaussian upper and lower bounds on the fundamental solution (heat kernel) of the operator, analogous to those for the pure heat equation.^[11] In the special case of the heat equation \partial_t u = \Delta u on \mathbb{R}^n \times (0, \infty), the forward Harnack inequality admits an explicit form derived directly from the Gaussian heat kernel K(t, x, y) = (4\pi t)^{-n/2} \exp(-|x - y|^2 / (4t)). For non-negative solutions and $0 < t < s, one has

\frac{u(x, t)}{u(y, s)} \leq \left( \frac{s}{t} \right)^{n/2} \exp\left( \frac{|x - y|^2}{4(s - t)} \right),

where the exponential term captures the diffusive spread over the time difference s - t. This precise estimate follows from the representation formula for solutions and the positivity of the kernel.

Applications and Extensions

Regularity and Liouville Theorems

Harnack's principle provides a key tool for establishing regularity properties of solutions to elliptic partial differential equations (PDEs). For a sequence of non-negative solutions to an elliptic equation that converge pointwise to a limit function on a domain, the principle asserts that the convergence is uniform on compact subsets, implying that the limit is also a solution and inherits higher regularity.^[8] This uniform convergence on compacts ensures that local boundedness and continuity propagate through the sequence, facilitating the proof of interior regularity estimates.^[13] A fundamental application of Harnack's inequality is the Liouville theorem for harmonic functions, which states that any bounded harmonic function defined on all of \mathbb{R}^n must be constant. The proof applies Harnack's inequality to balls of large radius R centered at the origin containing a fixed point x; as R \to \infty with r = |x|, the bounding factors \left( \frac{R - r}{R + r} \right)^{n-1} and \left( \frac{R + r}{R - r} \right)^{n-1} both approach 1, implying u(x) = u(0) for all x, hence u is constant. This contradicts the assumption of non-constancy under boundedness.^[3] This result, originally due to Liouville in the complex analytic setting and extended to harmonic functions, underscores the rigidity of entire solutions in Euclidean space.^[14] For more general elliptic operators, a Liouville-type theorem holds for positive solutions to the equation Lu = 0 in \mathbb{R}^n, where L is a uniformly elliptic operator of the form L u = a_{ij} \partial_i \partial_j u + b_i \partial_i u + c u with c \leq 0. Such positive solutions must be constant, as Harnack's inequality applied to growing balls forces the gradient to vanish everywhere, leveraging the non-positivity of the zeroth-order term to prevent growth. This extension highlights the role of Harnack estimates in controlling the behavior of subsolutions without lower-order growth terms. In the parabolic setting, a analogous Liouville theorem applies to ancient bounded solutions of the heat equation \partial_t u = \Delta u on \mathbb{R}^n \times (-\infty, T), asserting that they must be constant in space and time. By rescaling time and applying parabolic Harnack inequalities on expanding cylinders, the boundedness implies spatial homogeneity, reducing the solution to a constant.^[15] This result, building on classical work by Bernstein and Pini, captures the asymptotic stability of the heat flow for entire solutions defined backward in time.^[16] Harnack's inequality plays a pivotal role in the De Giorgi-Nash-Moser theory, which establishes higher regularity for weak solutions to uniformly elliptic and parabolic equations with bounded measurable coefficients. Specifically, the weak Harnack inequality for non-negative subsolutions implies Hölder continuity via Moser's iteration technique, where reverse Hölder estimates bound the L^p-norms and yield oscillation decay, culminating in C^{0,\alpha} estimates independent of the solution's size.^[13] This framework, initiated by De Giorgi in 1957, Nash in 1958, and Moser in 1960, revolutionized the regularity theory by bypassing classical C^2 assumptions. An illustrative application arises in the Bernstein theorem for minimal surfaces, where entire graphs minimizing area in \mathbb{R}^3 are planes, proved using elliptic Harnack inequalities for the associated quasilinear equation. For a graph u: \mathbb{R}^2 \to \mathbb{R} satisfying the minimal surface equation \operatorname{div} \left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = 0, Harnack estimates on positive auxiliary functions imply bounded gradients, forcing u to be affine.^[17] This result, extended to higher dimensions up to n=7 by Bombieri-De Giorgi-Giusti, relies on the uniform ellipticity preserved by Harnack principles for the nonlinear operator.^[18]

Use in Geometric Analysis

Harnack's inequality plays a pivotal role in geometric analysis, particularly in the study of evolving Riemannian geometries under flows like the Ricci flow, where it provides bounds on curvature evolution. In 1993, Richard Hamilton established a differential Harnack estimate for the Ricci flow, which quantifies the evolution of the curvature tensor along the flow. Specifically, for ancient solutions—those defined for all negative times—the estimate bounds the norm of the Riemann curvature tensor by |\mathrm{Rm}|(t) \leq C / t, where C is a constant depending on initial data, thereby controlling the growth of curvature as time approaches singularities and enabling analysis of long-time behavior.^[19] This estimate has been instrumental in understanding the qualitative dynamics of Ricci flow on compact manifolds. Building on Hamilton's work, Grigory Perelman extended Harnack-type inequalities through his development of entropy functionals in 2002–2003, which are monotone under the Ricci flow and incorporate differential Harnack expressions to measure entropy decrease. These functionals, combined with Harnack estimates adapted to Ricci flow with surgery, allowed Perelman to resolve the Poincaré conjecture by demonstrating that the flow on simply connected three-manifolds with positive Ricci curvature converges to a round metric after finite-time singularities are surgically removed.^[20] The Harnack inequalities ensured control over curvature near surgery caps and in the evolving topology, facilitating the decomposition into prime pieces and proving the conjecture's topological implications.^[21] In mean curvature flow, Harnack-type estimates similarly bound the evolution of mean curvature for hypersurfaces. Hamilton's 1995 Harnack estimate for the mean curvature flow provides differential inequalities that control curvature gradients, and these have been applied to graphical mean curvature flows, where the hypersurface is represented as a graph over a domain. Such estimates ensure long-time existence and regularity for graphical flows in Euclidean space, preventing singularities by bounding curvature decay and enabling uniqueness results for entire graphs. For Kähler-Ricci flow, parabolic Harnack inequalities preserve the Kähler structure along the evolution. In 1992, Huai-Dong Cao derived Harnack estimates tailored to the Kähler-Ricci flow, showing that positive initial Kähler metrics remain Kähler under the flow while bounding the potential function's gradients and oscillations. These inequalities facilitate convergence to Kähler-Einstein metrics on Fano manifolds with positive anticanonical bundles, maintaining the complex structure throughout the parabolic evolution. Modern extensions of Harnack inequalities to general Riemannian manifolds incorporate geometric invariants like injectivity radius and Ricci curvature bounds. For instance, on complete manifolds with Ricci curvature bounded below and positive injectivity radius, local Harnack estimates for positive harmonic functions depend on the injectivity radius i(p) and Ricci lower bound \mathrm{Ric} \geq -(n-1)K, yielding oscillation controls of the form \sup_B u / \inf_B u \leq C(n, K, r/i(p)), where B is a geodesic ball of radius r. These bounds enhance volume comparisons and gradient estimates, crucial for studying manifold geometry without compactness assumptions.^[22] A notable application appears in the Brendle-Schoen theorem on manifolds with positive isotropic curvature. In their 2008–2010 works, Simon Brendle and Richard Schoen proved that compact simply connected n-manifolds (n \geq 4) with positive isotropic curvature are diffeomorphic to the standard sphere by establishing that the condition is preserved under Ricci flow and deriving curvature pinching along the flow.^[23]