Harmonic functions are the backbone of complex analysis, bridging the gap between real and imaginary parts of analytic functions. They're smooth, satisfy Laplace's equation, and have no local extrema inside their domain. These properties make them crucial in physics and engineering.

The mean value property and maximum principle are key features of harmonic functions. They tell us that a harmonic function's value at a point equals the average on a surrounding sphere, and that it can't have internal maxima or minima. This leads to unique solutions for boundary value problems.

Harmonic functions and their properties

Definition and key characteristics

• A harmonic function is a twice continuously differentiable, real-valued function that satisfies Laplace's equation: $\nabla^2u = 0$, where $\nabla^2$ is the Laplace operator
• Harmonic functions are infinitely differentiable (smooth) and analytic, meaning they can be represented by a convergent power series in a neighborhood of every point in their domain
• The sum, difference, and constant multiples of harmonic functions are also harmonic, making the set of harmonic functions a vector space over the real numbers
• If $f(z)$ is an analytic function, then both its real and imaginary parts, $\Re(f(z))$ and $\Im(f(z))$, are harmonic functions

Properties and implications

• Harmonic functions have no local maxima or minima within their domain, except at boundary points or points where the function is constant
  • This property is a consequence of the maximum principle, which states that a non-constant harmonic function cannot attain its maximum or minimum value within its domain
• Harmonic functions satisfy the mean value property, which states that the value of a harmonic function at the center of a ball is equal to the average of its values on the surface of the ball
• Harmonic functions are uniquely determined by their boundary values on a bounded domain, as stated by the uniqueness theorem for the Dirichlet problem
• The properties of harmonic functions make them useful in various applications, such as in the study of electrostatics, fluid dynamics, and heat conduction

Mean value property for harmonic functions

Statement and formulation

• The mean value property states that the value of a harmonic function at the center of a ball is equal to the average of its values on the surface of the ball
• For a harmonic function $u(x)$ defined on a ball $B(x_0, r)$ with center $x_0$ and radius $r$, the mean value property is expressed as: $u(x_0) = \frac{1}{|\partial B|} \int_{\partial B} u(x_0 + r\xi) , dS(\xi)$ where $|\partial B|$ is the surface area of the ball and $\xi$ is a unit vector

Proof and implications

• The proof of the mean value property involves using Green's identities and the divergence theorem to transform the surface integral into a volume integral
  • The volume integral simplifies to the value of the function at the center due to the harmonicity of $u$, as $\nabla^2u = 0$
• The mean value property is a characteristic property of harmonic functions and can be used to prove other properties, such as the maximum principle and Harnack's inequality
• The mean value property also has practical applications, such as in the numerical solution of Laplace's equation using finite difference methods

Maximum principle for harmonic functions

Statement and consequences

• The maximum principle states that if a harmonic function attains its maximum or minimum value within its domain, then the function must be constant throughout the domain
• A consequence of the maximum principle is that a non-constant harmonic function cannot attain its maximum or minimum value within its domain; extrema can only occur on the boundary of the domain
• The strong maximum principle states that if two harmonic functions agree at an interior point of a connected domain and one dominates the other on the domain, then the functions are identical throughout the domain

Applications and extensions

• The maximum principle is useful for proving uniqueness theorems and for deriving estimates and bounds for harmonic functions
  • For example, the maximum principle can be used to show that a harmonic function is bounded by its maximum and minimum values on the boundary of its domain
• The maximum principle can be extended to subharmonic and superharmonic functions, which are functions that satisfy inequalities related to the Laplace operator
• The maximum principle also has analogues for other elliptic partial differential equations, such as the heat equation and the Schrödinger equation

Uniqueness of harmonic functions with boundary conditions

Dirichlet problem and uniqueness theorem

• The Dirichlet problem seeks to find a harmonic function on a domain that satisfies given boundary conditions
• The uniqueness theorem for the Dirichlet problem states that if a harmonic function satisfies given continuous boundary conditions on a bounded domain, then it is the unique solution to the Dirichlet problem

Proof and extensions

• The proof of uniqueness relies on the maximum principle: if two harmonic functions satisfy the same boundary conditions, their difference is a harmonic function that vanishes on the boundary, implying that the difference is identically zero throughout the domain
• Uniqueness results can be extended to unbounded domains under suitable growth conditions on the harmonic functions, such as requiring the functions to be bounded or have a certain asymptotic behavior (e.g., vanishing at infinity)
• Uniqueness theorems also hold for other boundary value problems, such as the Neumann problem, which specifies the normal derivative of the function on the boundary instead of the function values themselves