8.1 Harnack's inequality

Harnack's inequality is a key result in potential theory, providing quantitative estimates for positive harmonic and superharmonic functions. It establishes a relationship between maximum and minimum values on compact subsets of a domain, offering crucial insights into function behavior.

This inequality has far-reaching consequences, including Harnack's principle, Liouville's theorem, and Hölder continuity of harmonic functions. It's been generalized to various settings, including Riemannian manifolds and elliptic operators, expanding its applicability in potential theory and PDEs.

Definition of Harnack's inequality

• Fundamental result in the theory of harmonic and superharmonic functions
• Provides a quantitative estimate of the oscillation of a positive harmonic or superharmonic function on a domain
• Establishes a relationship between the maximum and minimum values of a function on a compact subset of the domain

Harnack's inequality for harmonic functions

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• Applies to positive harmonic functions $u$ on a domain $\Omega$
• States that for any compact subset $K \subset \Omega$, there exists a constant $C > 0$ such that $\max_K u \leq C \min_K u$
• The constant $C$ depends on the dimension, the domain, and the distance between $K$ and $\partial \Omega$
• Example: If $u$ is a positive harmonic function on the unit ball $B(0, 1)$, then $\max_{B(0, 1/2)} u \leq 3^n \min_{B(0, 1/2)} u$, where $n$ is the dimension

Harnack's inequality for superharmonic functions

• Extends Harnack's inequality to positive superharmonic functions
• A function $u$ is superharmonic if $-u$ is subharmonic, meaning $u$ satisfies the mean value inequality: $u(x) \geq \frac{1}{|B(x, r)|} \int_{B(x, r)} u(y) dy$ for all balls $B(x, r) \subset \Omega$
• Harnack's inequality for superharmonic functions states that for any compact subset $K \subset \Omega$, there exists a constant $C > 0$ such that $\max_K u \leq C u(x)$ for all $x \in K$

Constants in Harnack's inequality

• The constant $C$ in Harnack's inequality depends on several factors:
  • The dimension of the space
  • The geometry of the domain $\Omega$
  • The distance between the compact set $K$ and the boundary $\partial \Omega$
• In some cases, explicit constants can be obtained, such as the $3^n$ constant for the unit ball in $\mathbb{R}^n$
• Finding sharp constants in Harnack's inequality is an active area of research

Consequences of Harnack's inequality

• Harnack's inequality has several important consequences in potential theory and the study of elliptic partial differential equations
• Provides a powerful tool for studying the behavior of harmonic and superharmonic functions

Harnack's principle

• A qualitative version of Harnack's inequality
• States that if a sequence of positive harmonic functions converges at a single point, then it converges uniformly on compact subsets of the domain
• Useful for proving the existence and uniqueness of solutions to various boundary value problems

Liouville's theorem

• A consequence of Harnack's inequality for harmonic functions on the entire space $\mathbb{R}^n$
• States that any positive harmonic function on $\mathbb{R}^n$ must be constant
• Demonstrates the rigidity of harmonic functions on unbounded domains

Hölder continuity of harmonic functions

• Harnack's inequality implies that harmonic functions are locally Hölder continuous
• For any compact subset $K \subset \Omega$, there exist constants $C > 0$ and $\alpha \in (0, 1)$ such that $|u(x) - u(y)| \leq C |x - y|^\alpha$ for all $x, y \in K$
• The Hölder exponent $\alpha$ depends on the dimension and the distance between $K$ and $\partial \Omega$

Proof of Harnack's inequality

• The proof of Harnack's inequality relies on several key techniques and inequalities in potential theory
• Different approaches can be used depending on the context and the desired level of generality

Poisson kernel representation

• Expresses a positive harmonic function $u$ on a ball $B(x, r)$ as an integral of its boundary values against the Poisson kernel
• The Poisson kernel is given by $P(x, y) = \frac{r^2 - |x - y|^2}{n \alpha(n) r |x - y|^n}$ for $x \in B(x, r)$ and $y \in \partial B(x, r)$, where $\alpha(n)$ is the volume of the unit ball in $\mathbb{R}^n$
• Allows for estimating the values of $u$ inside the ball in terms of its boundary values

Harnack chains

• A technique for comparing the values of a positive harmonic function at two points in a domain
• Constructs a chain of balls connecting the two points, such that the ratio of the function values on consecutive balls is controlled by Harnack's inequality
• The number of balls in the chain depends on the distance between the points and the geometry of the domain

Caccioppoli inequality

• A key ingredient in the proof of Harnack's inequality
• Provides an estimate for the $L^2$ norm of the gradient of a harmonic function in terms of its $L^2$ norm on a larger set
• Specifically, if $u$ is harmonic on $B(x, r)$, then $\int_{B(x, r/2)} |\nabla u|^2 dx \leq \frac{C}{r^2} \int_{B(x, r)} u^2 dx$ for some constant $C > 0$

Moser's iteration technique

• A powerful method for deriving Harnack's inequality from the Caccioppoli inequality
• Involves iteratively applying the Caccioppoli inequality to obtain $L^p$ estimates for the function with increasing values of $p$
• Leads to a bound on the supremum of the function in terms of its $L^p$ norm, which can be translated into Harnack's inequality

Generalizations of Harnack's inequality

• Harnack's inequality has been generalized to various settings beyond harmonic functions on Euclidean domains
• These generalizations extend the applicability of the inequality to a wider range of problems in potential theory and PDEs

Harnack's inequality on Riemannian manifolds

• Extends Harnack's inequality to positive harmonic functions on Riemannian manifolds
• Requires the manifold to satisfy certain geometric conditions, such as non-negative Ricci curvature or a doubling property for the volume of balls
• The constant in the inequality depends on the geometry of the manifold, such as the injectivity radius and the curvature bounds

Harnack's inequality for elliptic operators

• Generalizes Harnack's inequality to positive solutions of elliptic partial differential equations
• Considers operators of the form $Lu = -\text{div}(A(x) \nabla u) + b(x) \cdot \nabla u + c(x) u$, where $A$ is a uniformly positive definite matrix, and $b$ and $c$ are bounded coefficients
• The constants in the inequality depend on the ellipticity of the operator and the regularity of the coefficients

Harnack's inequality for parabolic equations

• Adapts Harnack's inequality to positive solutions of parabolic partial differential equations, such as the heat equation
• Involves comparing the values of the solution at different times and locations
• The inequality takes the form $u(x, t) \leq C u(y, s)$ for $(x, t)$ and $(y, s)$ satisfying certain space-time conditions
• The constant $C$ depends on the parabolic operator and the space-time geometry

Applications of Harnack's inequality

• Harnack's inequality and its generalizations have numerous applications in potential theory, PDEs, and other areas of analysis
• Provides a powerful tool for studying the behavior of solutions to various problems

Boundary Harnack principle

• An extension of Harnack's inequality that compares the values of positive harmonic functions near the boundary of a domain
• Useful for studying the boundary behavior of solutions to elliptic boundary value problems
• Plays a crucial role in the study of the Martin boundary and the construction of the Martin kernel

Regularity of solutions to elliptic PDEs

• Harnack's inequality can be used to derive regularity estimates for solutions to elliptic PDEs
• Implies that solutions are locally Hölder continuous and can be used to establish higher-order regularity properties
• Helps in understanding the smoothness of solutions and their dependence on the data and the coefficients of the equation

Convergence of solutions to elliptic PDEs

• Harnack's inequality is a key tool in proving the convergence of sequences of solutions to elliptic PDEs
• Allows for obtaining uniform estimates on the solutions and their derivatives
• Used in the study of homogenization, singular perturbations, and other asymptotic problems in PDEs

Harnack's inequality in potential theory

• Harnack's inequality is a fundamental result in potential theory, which studies the properties of harmonic and subharmonic functions
• Provides a quantitative estimate of the oscillation of potentials and Green's functions
• Plays a role in the study of capacity, polar sets, and fine properties of potentials
• Used in the construction of the Martin boundary and the study of minimal positive harmonic functions