Subharmonic Functions

Definition

Subharmonic functions are real-valued functions that are upper semi-continuous and satisfy the mean value property in a certain sense. These functions are closely related to harmonic functions, as they can be thought of as functions that lie below harmonic functions in a way that they do not exceed the average of their values on any surrounding sphere, making them important in potential theory and analysis.

5 Must Know Facts For Your Next Test

1. Subharmonic functions are defined on an open set and can take infinite values at points where they are not defined.
2. These functions can be thought of as 'non-positive' solutions to certain differential inequalities, which distinguish them from harmonic functions.
3. They can serve as useful barriers for establishing properties of harmonic functions, especially when using techniques like the maximum principle.
4. In many scenarios, subharmonic functions may not be differentiable but still maintain properties that make them significant in potential theory.
5. The concept of subharmonicity is crucial when applying Green's identities, as it helps identify relationships between various potential functions.

Review Questions

• How do subharmonic functions relate to harmonic functions in terms of their mean value properties?
  • Subharmonic functions are essentially the opposite of harmonic functions when it comes to the mean value property. While harmonic functions have values equal to the average of their values over spheres surrounding a point, subharmonic functions do not exceed this average. This relationship indicates that subharmonic functions lie below harmonic ones and can provide insights into the behavior and regularity of solutions to potential theory problems.
• Discuss the significance of upper semi-continuity in the context of subharmonic functions and how it influences their behavior.
  • Upper semi-continuity plays a critical role in defining subharmonic functions since it ensures that these functions do not 'jump' up at any point. This property allows subharmonic functions to be well-behaved under limits, which is essential when considering convergence properties or applying maximum principles. The upper semi-continuous nature also supports the idea that subharmonic functions can serve as effective barriers for harmonic ones, leading to useful applications in various mathematical scenarios.
• Evaluate how the concept of subharmonic functions aids in understanding Green's identities and their implications for potential theory.
  • Subharmonic functions significantly enhance our understanding of Green's identities by providing a framework for relating different potentials. Green's identities establish relationships between the values of subharmonic and harmonic functions, facilitating the application of boundary conditions and integral formulations. By using subharmonic functions as barriers, one can derive estimates for harmonic functions, thereby gaining insights into their regularity and behavior within specific domains. This interplay between these concepts is essential for solving complex problems in potential theory.

Related terms

Harmonic Functions: Functions that are twice continuously differentiable and satisfy Laplace's equation, which means they have a mean value property that averages values over spheres.

Upper Semi-Continuity: A property of a function where the limit of the function values at a point is less than or equal to the function value at that point.

Superharmonic Functions:

Functions that are lower semi-continuous and satisfy the mean value property in the reverse sense, meaning they are greater than or equal to the average of their values on any surrounding sphere.