Subharmonic Functions

Definition

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.

5 Must Know Facts For Your Next Test

1. Subharmonic functions are locally integrable and can be considered as lower semi-continuous functions, which allows them to be analyzed using similar techniques as harmonic functions.
2. They do not have to be differentiable, but if they are, they satisfy the Laplacian inequality $$ abla^2 u \geq 0$$ at points where they are twice differentiable.
3. Subharmonic functions can approach infinity on their domain, making them useful in various applications such as potential theory and complex analysis.
4. The subharmonicity condition allows for the use of comparison principles with harmonic functions, providing tools for analyzing convergence and continuity properties.
5. Every harmonic function is subharmonic, but not every subharmonic function is harmonic; this relationship highlights their role in generalizing properties of harmonic functions.

Review Questions

• How do subharmonic functions differ from harmonic functions in terms of their defining properties?
  • Subharmonic functions differ from harmonic functions primarily in that they satisfy a weaker form of the mean value property. While harmonic functions have the value at any point equal to the average over surrounding spheres, subharmonic functions only require that their value is less than or equal to this average. This means that while all harmonic functions are subharmonic, subharmonic functions may not exhibit all the properties of harmonic ones.
• Discuss how subharmonic functions can be utilized to study the properties of harmonic functions through comparison principles.
  • Subharmonic functions play an important role in studying harmonic functions by allowing for comparison principles. When we analyze a subharmonic function alongside a harmonic function, we can draw conclusions about limits and behavior of both types of functions. For instance, if we know a certain region contains both types of functions, we can often infer properties about the harmonic function by using bounds provided by the subharmonic function, aiding in solving various problems in complex analysis and potential theory.
• Evaluate the implications of subharmonicity on the behavior of a function near its boundary, particularly in relation to maximum principles.
  • The implications of subharmonicity on a function's behavior near its boundary can be significant due to maximum principles. Since a subharmonic function cannot attain its maximum value inside its domain unless it is constant, this characteristic ensures that values will tend to drop as one moves inward from the boundary. This leads to important conclusions about the existence of extremal values and contributes to understanding how solutions behave near boundaries in various mathematical contexts. Such principles can be crucial for establishing uniqueness in solutions to boundary-value problems.

Related terms

Harmonic Functions: Functions that satisfy Laplace's equation and exhibit the mean value property, meaning their value at a point equals the average of their values over any sphere surrounding that point.

Superharmonic Functions: Functions that are the opposite of subharmonic functions, where at every point in their domain, their value is greater than or equal to the average value over any sphere centered at that point.

Laplace's Equation: A second-order partial differential equation given by $$ abla^2 u = 0$$, which harmonic functions satisfy, playing a crucial role in defining the behavior of subharmonic and superharmonic functions.