Glossary

9.1 Properties of harmonic functions

4 min read•august 13, 2024

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

The and 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

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  • A harmonic function is a twice continuously differentiable, real-valued function that satisfies Laplace's equation: ∇2u=0\nabla^2u = 0, where ∇2\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)f(z) is an analytic function, then both its real and imaginary parts, ℜ(f(z))\Re(f(z)) and ℑ(f(z))\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
  • 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)u(x) defined on a ball B(x0,r)B(x_0, r) with center x0x_0 and radius rr, the mean value property is expressed as: u(x0)=1∣∂B∣∫∂Bu(x0+rξ) dS(ξ)u(x_0) = \frac{1}{|\partial B|} \int_{\partial B} u(x_0 + r\xi) \, dS(\xi) where ∣∂B∣|\partial B| is the surface area of the ball and ξ\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 uu, as ∇2u=0\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
  • 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 , 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 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

Key Terms to Review (16)

Analytic continuation: Analytic continuation is a technique in complex analysis that extends the domain of a given analytic function beyond its original radius of convergence. This method allows for the function to be expressed in terms of another analytic function, effectively 'continuing' it in a larger region. It connects deeply with concepts like singularities, branch points, and the behavior of functions across different domains.
Boundary Value Problem: A boundary value problem involves finding a function that satisfies a differential equation along with specific conditions imposed on the function at the boundaries of its domain. This type of problem is crucial in understanding physical systems described by differential equations, as it helps to determine solutions that behave well at the edges of the region of interest, ensuring the function meets required criteria in real-world applications.
Dirichlet Problem: The Dirichlet Problem is a type of boundary value problem where the goal is to find a harmonic function defined in a domain, given the values that the function must take on the boundary of that domain. This problem is fundamental in the study of harmonic functions and their properties, and is closely linked to various methods for finding solutions, such as the Poisson integral formula and Green's functions, which can be employed to tackle these kinds of problems effectively.
Harmonic conjugate: A harmonic conjugate is a function that, when paired with a given harmonic function, forms a complex analytic function. In other words, if u(x,y) is a harmonic function, then its harmonic conjugate v(x,y) satisfies the Cauchy-Riemann equations, making the complex function f(z) = u(x,y) + iv(x,y) differentiable. This relationship illustrates the deep connection between harmonic functions and complex analysis, showcasing how these concepts can work together to understand the behavior of complex functions.
Harnack's Inequality: Harnack's Inequality is a fundamental result in the study of harmonic functions that provides a relationship between the values of a positive harmonic function at different points in a domain. Specifically, it states that if a harmonic function is positive on a connected open set, then there exists a constant such that the values of the function at any two points in that set are comparable. This inequality is crucial for establishing the continuity and regularity properties of harmonic functions.
Heat equation: The heat equation is a partial differential equation that describes how heat diffuses through a given region over time. It provides a mathematical framework to understand the distribution of temperature in a medium, making it fundamental in studying harmonic functions as it exhibits properties such as smoothness and the mean value property.
Isolated Singularity: An isolated singularity is a point in the complex plane where a function ceases to be analytic, but is analytic in some neighborhood around that point, except for the singularity itself. This concept is crucial as it helps in understanding the behavior of complex functions near points where they may not be well-defined, and it connects to various important results and tools in complex analysis.
Laplace's equation: Laplace's equation is a second-order partial differential equation given by $$ abla^2 u = 0$$, where $$u$$ is a scalar function and $$ abla^2$$ is the Laplacian operator. This equation plays a crucial role in understanding harmonic functions, which are solutions to this equation, and it helps to describe various physical phenomena such as heat conduction and fluid flow. Its properties and applications extend to boundary value problems, making it essential in fields like physics and engineering.
Maximum Principle: The maximum principle states that if a function is harmonic on a connected open set, then its maximum value occurs on the boundary of that set. This principle highlights the behavior of harmonic functions and is crucial in understanding their properties and implications in various contexts such as potential theory and boundary value problems.
Mean Value Property: The mean value property states that for a harmonic function defined on a domain, the value at any point is equal to the average of its values over any surrounding sphere. This concept is fundamental in understanding the behavior of harmonic functions and connects deeply to the properties that define them, as well as their solutions in boundary value problems and representations through integral formulas.
Poisson's Equation: Poisson's Equation is a fundamental partial differential equation of the form $$ abla^2 u = f$$, where $$u$$ is the unknown function and $$f$$ is a known function representing sources or sinks. This equation plays a crucial role in various fields, including physics and engineering, particularly in the study of electrostatics, heat conduction, and fluid dynamics, as it relates to harmonic functions and their properties. Understanding this equation provides insights into how potential functions behave under different conditions.
Potential Theory: Potential theory is a branch of mathematical analysis that studies harmonic functions and their properties, particularly in relation to the concepts of potential energy and fields. It focuses on functions that satisfy Laplace's equation, which are crucial for understanding physical phenomena in various fields like fluid dynamics and electrostatics. This theory connects deeply with harmonic functions, as they represent potential fields, and it finds practical applications in areas like physics and engineering, where these concepts help model real-world problems.
Removable singularity: A removable singularity is a type of isolated singularity where a function can be defined at that point so that it becomes analytic there. This means that if a function has a removable singularity, it can be 'fixed' by redefining it at that point, making it continuous and differentiable in the neighborhood around it. This concept relates to how functions behave near points where they seem undefined or behave poorly, showing the underlying structure of analytic functions.
Riesz Representation Theorem: The Riesz Representation Theorem establishes a powerful connection between harmonic functions and measures, stating that every bounded linear functional on a space of continuous functions can be represented as an integral with respect to a unique positive Borel measure. This theorem highlights the relationship between harmonic functions, which are solutions to Laplace's equation, and the properties of these functions through the use of integrals, ultimately leading to deeper insights into potential theory and function spaces.
Subharmonic Functions: Subharmonic functions are real-valued functions defined on a domain that satisfy the mean value property for harmonic functions in a weakened form. Specifically, a function is subharmonic if, at every point in its domain, its value is less than or equal to the average value of the function over any sphere centered at that point. This concept connects closely with the properties of harmonic functions, as subharmonic functions can be viewed as generalizations that exhibit certain analogous behaviors.
Superharmonic Functions: Superharmonic functions are real-valued functions that are upper semi-continuous and satisfy the mean value property for any ball in their domain, meaning that the function's value at any point is greater than or equal to the average value over any surrounding ball. These functions generalize harmonic functions and exhibit important properties, such as being associated with subharmonic functions. Superharmonic functions play a key role in potential theory and variational problems.
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