1.2 Harmonic functions

Harmonic functions are twice continuously differentiable functions that satisfy Laplace's equation. They play a key role in potential theory, appearing in electrostatics, fluid dynamics, and heat conduction. These functions have unique properties that make them essential in mathematical physics.

The mean value property and maximum principle are fundamental characteristics of harmonic functions. These properties allow us to understand their behavior and solve boundary value problems. Harnack's inequality and the Poisson integral formula provide powerful tools for analyzing and constructing harmonic functions in various domains.

Definition of harmonic functions

• Harmonic functions are twice continuously differentiable functions that satisfy Laplace's equation
• Play a fundamental role in potential theory, a branch of mathematics that studies the behavior of functions satisfying certain partial differential equations
• Arise naturally in various contexts, such as in the study of electrostatics, fluid dynamics, and heat conduction

Laplace's equation

Solutions in various dimensions

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• In one dimension, Laplace's equation reduces to $\frac{d^2u}{dx^2} = 0$, and its solutions are linear functions of the form $u(x) = ax + b$
• In two dimensions, Laplace's equation is $\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$, and its solutions include harmonic polynomials and logarithmic functions
• In three dimensions, Laplace's equation is $\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2} = 0$, and its solutions include spherical harmonics and Coulomb potentials
• Solutions to Laplace's equation in higher dimensions have applications in quantum mechanics and string theory

Mean value property

Integral formulas

• The mean value property states that the value of a harmonic function at any point is equal to the average of its values over any sphere or ball centered at that point
• For a harmonic function $u$ defined on a domain $\Omega \subset \mathbb{R}^n$, the mean value property can be expressed as $u(x) = \frac{1}{|\partial B(x, r)|} \int_{\partial B(x, r)} u(y) \, dS(y)$, where $B(x, r)$ is a ball of radius $r$ centered at $x$
• The mean value property can also be expressed using volume integrals: $u(x) = \frac{1}{|B(x, r)|} \int_{B(x, r)} u(y) \, dy$
• These integral formulas provide a way to reconstruct a harmonic function from its boundary values

Maximum principle

Consequences and applications

• The maximum principle states that a non-constant harmonic function cannot attain its maximum or minimum value at an interior point of its domain
• A consequence of the maximum principle is that if two harmonic functions agree on the boundary of a domain, they must agree everywhere inside the domain
• The maximum principle is useful in proving uniqueness results for boundary value problems involving Laplace's equation
• The maximum principle has applications in the study of elliptic partial differential equations and in the theory of Markov processes

Harnack's inequality

Harnack's convergence theorem

• Harnack's inequality provides an estimate for the ratio of the values of a positive harmonic function at two points in terms of the distance between the points and their distance to the boundary of the domain
• For a positive harmonic function $u$ defined on a domain $\Omega \subset \mathbb{R}^n$, Harnack's inequality states that $\frac{u(x)}{u(y)} \leq C \left(\frac{|x-y|}{d(x, \partial \Omega)}\right)^{\alpha}$, where $C$ and $\alpha$ are positive constants that depend only on the dimension $n$
• Harnack's convergence theorem states that a bounded sequence of harmonic functions on a domain has a subsequence that converges uniformly on compact subsets to a harmonic function
• Harnack's inequality and convergence theorem are powerful tools in the study of the boundary behavior of harmonic functions and in the development of a potential theory for more general elliptic operators

Poisson integral formula

Dirichlet problem for a disk

• The Poisson integral formula provides a solution to the Dirichlet problem for Laplace's equation on a disk in terms of the boundary values of the function
• For a continuous function $f$ defined on the unit circle $\partial D$, the Poisson integral of $f$ is the function $u(r, \theta) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1-r^2}{1-2r\cos(\theta-t)+r^2} f(e^{it}) \, dt$, which is harmonic on the unit disk $D$ and satisfies $u(e^{i\theta}) = f(e^{i\theta})$ for all $\theta$
• The Dirichlet problem for a disk asks to find a harmonic function on the disk that takes prescribed values on the boundary circle
• The Poisson integral formula provides an explicit solution to the Dirichlet problem for a disk, demonstrating the existence and uniqueness of solutions to this boundary value problem

Green's functions

Fundamental solution of Laplace's equation

• Green's functions are a powerful tool in the study of linear partial differential equations, particularly in solving boundary value problems
• For Laplace's equation, the Green's function $G(x, y)$ is a function of two variables that satisfies $\Delta_x G(x, y) = \delta(x-y)$, where $\delta$ is the Dirac delta function
• The fundamental solution of Laplace's equation in $\mathbb{R}^n$ is given by $\Phi(x) = \begin{cases} -\frac{1}{2\pi} \log|x|, & n = 2 \\ \frac{1}{(n-2)\omega_n} |x|^{2-n}, & n \geq 3 \end{cases}$, where $\omega_n$ is the surface area of the unit sphere in $\mathbb{R}^n$
• Green's functions can be used to represent harmonic functions in terms of their boundary values, providing an alternative approach to the Poisson integral formula

Representation of harmonic functions

Poisson integral vs Green's function

• Harmonic functions on a domain can be represented using either the Poisson integral formula or Green's functions
• The Poisson integral formula expresses a harmonic function in terms of its boundary values, while the Green's function representation involves the values of the function and its normal derivative on the boundary
• For a harmonic function $u$ on a domain $\Omega$ with boundary $\partial \Omega$, the Green's function representation is given by $u(x) = \int_{\partial \Omega} \left(u(y) \frac{\partial G(x, y)}{\partial \nu} - G(x, y) \frac{\partial u(y)}{\partial \nu}\right) \, dS(y)$, where $\nu$ is the outward unit normal to $\partial \Omega$
• The choice between the Poisson integral and Green's function representation depends on the specific problem and the available information about the harmonic function and its boundary values

Harmonic conjugates

Cauchy-Riemann equations

• Harmonic conjugates are pairs of harmonic functions that are related by the Cauchy-Riemann equations
• If $u(x, y)$ is a harmonic function, then there exists a harmonic function $v(x, y)$, called the harmonic conjugate of $u$, such that $f(z) = u(x, y) + iv(x, y)$ is an analytic function of the complex variable $z = x + iy$
• The Cauchy-Riemann equations for a pair of harmonic functions $u$ and $v$ are given by $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• Harmonic conjugates play a crucial role in the connection between harmonic functions and complex analysis, allowing for the application of powerful tools from complex analysis to the study of harmonic functions

Subharmonic and superharmonic functions

Characterizations and properties

• Subharmonic functions are upper semicontinuous functions that satisfy the sub-mean value property, meaning that the value of the function at any point is less than or equal to the average of its values over any ball centered at that point
• Superharmonic functions are lower semicontinuous functions that satisfy the super-mean value property, with the inequality reversed
• Harmonic functions are both subharmonic and superharmonic
• The maximum principle holds for subharmonic functions: a subharmonic function cannot attain its maximum value at an interior point of its domain unless it is constant
• The minimum principle holds for superharmonic functions: a superharmonic function cannot attain its minimum value at an interior point of its domain unless it is constant
• Subharmonic and superharmonic functions arise in the study of more general elliptic partial differential equations and in the theory of viscosity solutions

Boundary behavior of harmonic functions

Fatou's theorem

• The boundary behavior of harmonic functions is a central topic in potential theory, as it relates to the solvability of boundary value problems and the regularity of solutions
• Fatou's theorem states that if $u$ is a positive harmonic function on the unit ball $B$ in $\mathbb{R}^n$, then $u$ has a non-tangential limit at almost every point of the boundary $\partial B$
• A non-tangential limit at a point $x \in \partial B$ is defined as the limit of $u(y)$ as $y$ approaches $x$ from within a cone with vertex at $x$ and aperture less than $\pi/2$
• Fatou's theorem has been generalized to harmonic functions on more general domains and to solutions of other elliptic partial differential equations

Liouville's theorem

Applications in complex analysis

• Liouville's theorem states that every bounded harmonic function on the entire space $\mathbb{R}^n$ must be constant
• In the context of complex analysis, Liouville's theorem implies that every bounded entire function (a function analytic on the entire complex plane) must be constant
• Liouville's theorem is a powerful result with many applications in complex analysis, such as in the proof of the fundamental theorem of algebra and in the classification of conformal mappings between domains
• Generalizations of Liouville's theorem, such as Picard's theorems, play a crucial role in the study of the behavior of analytic functions near singularities and in value distribution theory