Partial differential equations Parabolic equations Theorem

Harnack inequality for parabolic equations

Builds on Parabolic partial differential equation Weak solution Heat equation Harnack's inequality

Statement

Let $U \subseteq \R^n$ be open, let $T > 0$ , and let $0 < \lambda \le \Lambda$ . Let $A = (a_{ij})$ be a measurable function on $U \times (0, T]$ with values in the symmetric $n \times n$ matrices such that $\lambda \abs{\eta}^2 \le \sum_{i,j} a_{ij}(x, t)\, \eta_i \eta_j \le \Lambda \abs{\eta}^2$ for all $\eta \in \R^n$ and all $(x, t)$ . Let $u \ge 0$ be a weak solution of the parabolic equation $\partial_t u = \sum_{i,j=1}^{n} \partial_{x_i}\big(a_{ij}\, \partial_{x_j} u\big) \quad \text{in } U \times (0, T].$ Let $V$ be a connected open set with compact closure $\overline{V} \subset U$ , and let $0 < t_1 < t_2 \le T$ . Then there is a constant $C$ , depending only on $n$ , $\lambda$ , $\Lambda$ , $U$ , $V$ , $t_1$ and $t_2$ , such that $\sup_{x \in V} u(x, t_1) \le C \inf_{x \in V} u(x, t_2).$

Remarks

This is Moser's parabolic Harnack inequality of 1964; the solution is Hölder continuous by the De Giorgi–Nash–Moser theory, so the pointwise values make sense. Moser's proof bounds $\sup u$ and $\inf u$ on small cylinders by integral means of powers of $u$ , using iterated energy estimates. An estimate for $\log u$ then links the two bounds. A chain of overlapping cylinders then gives the statement on $V$ , which is why $V$ must be connected. The time lag $t_1 < t_2$ is necessary. For the heat equation $u_t = \Delta u$ on $\R^n$ , the solutions $u_y(x, t) = (4\pi t)^{-n/2} e^{-\abs{x - y}^2/(4t)}$ with $y \in \R^n$ satisfy $u_y(x_1, t)/u_y(x_2, t) = e^{(\abs{x_2 - y}^2 - \abs{x_1 - y}^2)/(4t)}$ , which is unbounded over $y$ for fixed distinct $x_1, x_2$ . So no bound $\sup_V u(\cdot, t) \le C \inf_V u(\cdot, t)$ at a single time can hold for all nonnegative solutions. The hypothesis $u \ge 0$ is also necessary, as for Harnack's inequality for harmonic functions.

Going deeper

For a student

The parabolic Harnack inequality is the time-dependent analogue of Harnack's inequality for nonnegative harmonic functions. It holds for equations with merely measurable coefficients, which makes it a central tool in the regularity theory of nonlinear parabolic equations. From it one derives Hölder continuity of solutions and a strong maximum principle. For the heat equation on Riemannian manifolds with nonnegative Ricci curvature, Li and Yau proved a sharp differential form, which integrates to a Harnack inequality with explicit constants.

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Euclidean space object
Function concept
Heat equation concept
Parabolic partial differential equation concept
Unit disc object
Weak solution concept
Total derivative concept
Function of several variables concept
Vector-valued function concept
Differentiable function of several variables concept
Partial derivative concept
Vector field concept
Gradient concept
Divergence concept
Laplacian concept
Harmonic function concept
Poisson kernel definition
Harnack's inequality statement
Harnack inequality for parabolic equations statement

Tag	`7BY5D2THKW`
URI	`m2://7BY5D2THKW`
Kind	statement, theorem
Writing	W0 stub
MSC	35K10, 35B65, 35B45
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About	Parabolic partial differential equation
Uses	Parabolic partial differential equation, Weak solution, Heat equation, Harnack's inequality
Changed	2 November 2026