8.2 Harnack's principle

Harnack's principle is a key concept in potential theory, linking values of non-negative harmonic functions across domains. It establishes that these functions can't have isolated zeros or sharp peaks, leading to important results in elliptic PDEs.

This principle has wide-ranging impacts, from the strong maximum principle to Harnack's convergence theorem. It's crucial for understanding boundary behavior of harmonic functions and solving the Dirichlet problem in potential theory.

Harnack's principle

• Harnack's principle is a fundamental result in the theory of harmonic functions and elliptic partial differential equations
• It establishes a relationship between the values of a non-negative harmonic function at different points in a domain
• Harnack's principle has far-reaching consequences in potential theory and the study of elliptic PDEs

Definition of Harnack's principle

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• States that if $u$ is a non-negative harmonic function in a domain $D$, then for any compact subset $K \subset D$, there exists a constant $C > 0$ such that $\max_{x \in K} u(x) \leq C \min_{x \in K} u(x)$
• The constant $C$ depends only on the domain $D$ and the compact subset $K$, but not on the specific function $u$
• Harnack's principle implies that non-negative harmonic functions cannot have isolated zeros or sharp peaks within a domain

Harnack's inequality

• A quantitative version of Harnack's principle that provides an explicit estimate for the constant $C$
• For a ball $B(x_0, r) \subset D$ and a non-negative harmonic function $u$ in $D$, Harnack's inequality states that $\sup_{B(x_0, r/2)} u \leq C \inf_{B(x_0, r/2)} u$, where $C$ depends only on the dimension and the radius $r$
• Harnack's inequality is a powerful tool for obtaining a priori estimates and regularity results for harmonic functions

Consequences of Harnack's principle

• Implies that non-negative harmonic functions satisfy the strong maximum principle: if $u$ attains its maximum in the interior of a domain, then $u$ must be constant
• Leads to the Harnack convergence theorem: a sequence of harmonic functions that is locally bounded above converges locally uniformly to a harmonic function
• Plays a crucial role in the study of the boundary behavior of harmonic functions and the Dirichlet problem

Harmonic functions

• Harmonic functions are twice continuously differentiable functions that satisfy Laplace's equation: $\Delta u = 0$, where $\Delta$ is the Laplace operator
• They appear naturally in various branches of mathematics and physics, such as potential theory, complex analysis, and electrostatics
• Harmonic functions have many remarkable properties, including the mean value property and the maximum principle

Definition of harmonic functions

• A function $u: D \to \mathbb{R}$ is harmonic in a domain $D \subset \mathbb{R}^n$ if it is twice continuously differentiable and satisfies Laplace's equation: $\sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2} = 0$
• Equivalently, $u$ is harmonic if and only if it satisfies the mean value property: for any ball $B(x, r) \subset D$, $u(x) = \frac{1}{|B(x, r)|} \int_{B(x, r)} u(y) dy$
• Examples of harmonic functions include linear functions, the real and imaginary parts of analytic functions, and the fundamental solution of Laplace's equation

Harnack's principle for harmonic functions

• Harnack's principle holds for non-negative harmonic functions in any domain $D \subset \mathbb{R}^n$
• It implies that the values of a non-negative harmonic function at any two points in a compact subset of $D$ are comparable up to a constant that depends only on the subset
• Harnack's principle is a key tool in the study of the boundary behavior and regularity of harmonic functions

Liouville's theorem

• A consequence of Harnack's principle for harmonic functions defined on the entire space $\mathbb{R}^n$
• Liouville's theorem states that any bounded harmonic function on $\mathbb{R}^n$ must be constant
• It demonstrates the rigidity of harmonic functions and the importance of boundary conditions in determining their behavior

Elliptic partial differential equations

• Elliptic PDEs are a class of second-order partial differential equations that generalize Laplace's equation
• They arise in various applications, such as elasticity theory, fluid dynamics, and quantum mechanics
• Harnack's principle and related techniques play a central role in the study of elliptic PDEs and their solutions

Laplace's equation

• The prototypical example of an elliptic PDE is Laplace's equation: $\Delta u = 0$, where $\Delta = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}$ is the Laplace operator
• Solutions to Laplace's equation are called harmonic functions and have numerous properties, such as the mean value property and the maximum principle
• Laplace's equation models steady-state heat conduction, electrostatics, and gravitational potential in various physical contexts

Poisson's equation

• A non-homogeneous version of Laplace's equation: $\Delta u = f$, where $f$ is a given function
• Poisson's equation describes the potential in the presence of a source or sink term $f$
• The fundamental solution of Poisson's equation is the Newtonian potential, which is used to construct solutions via the method of Green's functions

Harnack's principle for elliptic PDEs

• Harnack's principle extends to non-negative solutions of a wide class of elliptic PDEs, including those with variable coefficients and lower-order terms
• For a second-order elliptic PDE $Lu = 0$ with smooth coefficients, Harnack's principle states that non-negative solutions satisfy a Harnack inequality on compact subsets of the domain
• Harnack's principle for elliptic PDEs is a crucial tool for establishing regularity, maximum principles, and uniqueness results for their solutions

Applications of Harnack's principle

• Harnack's principle and its consequences have numerous applications in potential theory, PDEs, and geometric analysis
• They provide a powerful framework for studying the qualitative and quantitative properties of solutions to elliptic and parabolic equations
• Harnack's principle is also used in the investigation of harmonic manifolds and the geometry of Riemannian manifolds

Regularity of solutions

• Harnack's principle implies that solutions to elliptic and parabolic PDEs with smooth coefficients are locally Hölder continuous
• It allows for the derivation of a priori estimates on the modulus of continuity of solutions, which is crucial for proving existence and uniqueness results
• Harnack's principle is a key ingredient in the De Giorgi-Nash-Moser theory, which establishes the higher regularity of solutions to elliptic and parabolic PDEs with measurable coefficients

Maximum principles

• Harnack's principle is closely related to various maximum principles for elliptic and parabolic PDEs
• The weak maximum principle states that a subharmonic function (i.e., $\Delta u \geq 0$) in a domain $D$ attains its maximum on the boundary $\partial D$
• The strong maximum principle, a consequence of Harnack's principle, asserts that if a subharmonic function attains its maximum in the interior of $D$, then it must be constant
• Maximum principles are essential tools for comparing solutions, proving uniqueness, and analyzing the boundary behavior of solutions

Uniqueness of solutions

• Harnack's principle plays a crucial role in establishing the uniqueness of solutions to various boundary value problems for elliptic and parabolic PDEs
• For the Dirichlet problem for Laplace's equation, the maximum principle implies that the solution is unique
• In more general settings, Harnack's principle and maximum principles are combined with energy methods or the method of sub- and supersolutions to prove uniqueness results
• Uniqueness is a fundamental property that ensures the well-posedness of boundary value problems and enables the development of efficient numerical methods

Generalizations of Harnack's principle

• Harnack's principle has been extended and generalized to various settings beyond harmonic functions and elliptic PDEs
• These generalizations have found applications in the study of heat equations, minimal surfaces, and Riemannian manifolds
• The common theme in these generalizations is the control of the oscillation of solutions and the derivation of local estimates

Harnack's principle for parabolic PDEs

• Parabolic PDEs, such as the heat equation $\partial_t u - \Delta u = 0$, describe time-dependent diffusion processes
• Harnack's principle for parabolic PDEs states that non-negative solutions satisfy a Harnack inequality on compact subsets of the space-time domain
• The parabolic Harnack inequality is a crucial tool for establishing the regularity and asymptotic behavior of solutions to parabolic PDEs

Harnack's principle for subharmonic functions

• A function $u$ is subharmonic if it satisfies $\Delta u \geq 0$ in the distributional sense
• Harnack's principle for subharmonic functions states that if $u$ is a non-negative subharmonic function in a domain $D$, then for any compact subset $K \subset D$, there exists a constant $C > 0$ such that $\sup_{K} u \leq C \inf_{K} u$
• This generalization of Harnack's principle is used in the study of potential theory, pluripotential theory, and the theory of viscosity solutions

Harnack's principle in Riemannian geometry

• Harnack's principle has been extended to harmonic functions and elliptic PDEs on Riemannian manifolds
• For a Riemannian manifold $(M, g)$, a function $u$ is harmonic if it satisfies the Laplace-Beltrami equation $\Delta_g u = 0$, where $\Delta_g$ is the Laplace-Beltrami operator associated with the metric $g$
• Harnack's principle on Riemannian manifolds states that non-negative harmonic functions satisfy a Harnack inequality on compact subsets of the manifold, with the constant depending on the geometry of the manifold
• This generalization is a key tool in the study of harmonic manifolds, the geometry of Riemannian manifolds with non-negative Ricci curvature, and the analysis of heat kernels on manifolds