Subharmonic Functions

Definition

Subharmonic functions are real-valued functions defined on a domain in Euclidean space that satisfy the mean value property for all balls contained in that domain. This means that the value of a subharmonic function at any point is less than or equal to the average value of the function over any ball centered at that point. They are closely related to harmonic functions, which represent solutions to Laplace's equation, and they play an essential role in potential theory and complex analysis.

5 Must Know Facts For Your Next Test

1. Subharmonic functions are not necessarily continuous but must be upper semi-continuous.
2. The Laplacian of a subharmonic function is non-positive, meaning it can be zero or negative but not positive.
3. Any harmonic function is also subharmonic, since harmonic functions satisfy the mean value property strictly as an equality.
4. Subharmonic functions can be used to prove the existence of harmonic functions via the classical theory of potentials.
5. The maximum principle applies to subharmonic functions, indicating they cannot achieve a local maximum unless they are constant.

Review Questions

• How do subharmonic functions relate to harmonic functions and what properties do they share?
  • Subharmonic functions are closely related to harmonic functions, with every harmonic function being subharmonic. They share certain properties, such as the upper semi-continuity and the ability to have a non-positive Laplacian. However, unlike harmonic functions that meet the mean value property with equality, subharmonic functions meet it with an inequality. This relationship highlights their role in potential theory.
• Discuss the implications of the maximum principle for subharmonic functions and how it affects their behavior in a given domain.
  • The maximum principle for subharmonic functions states that they cannot achieve a local maximum within their domain unless they are constant. This means that if you have a subharmonic function defined on a bounded domain, its values will never exceed those at the boundary. This property is crucial in understanding the behavior of these functions and establishing control over their values in specific regions.
• Evaluate how subharmonic functions contribute to the understanding of potential theory and its applications.
  • Subharmonic functions play a significant role in potential theory by providing insights into the behavior of potentials associated with physical phenomena such as gravitational and electrostatic fields. Their properties help in proving the existence and uniqueness of harmonic potentials under certain conditions. Additionally, subharmonic functions are used in various applications ranging from complex analysis to mathematical physics, showcasing their importance in both theoretical and applied contexts.

Related terms

Harmonic Functions: Functions that satisfy Laplace's equation, meaning they have continuous second derivatives and their Laplacian equals zero. They exhibit nice properties like being infinitely differentiable.

Mean Value Property: The principle that for a harmonic function, the value at a point is equal to the average value over any surrounding ball. This property does not hold for subharmonic functions, where the inequality applies instead.

Superharmonic Functions: These are functions that are the opposite of subharmonic functions; at any point, a superharmonic function is greater than or equal to the average value over any ball centered at that point.