Superharmonic function

Definition

A superharmonic function is a function that is upper semicontinuous and satisfies the mean value property, meaning its value at any point is greater than or equal to the average of its values over any surrounding sphere. This concept plays a crucial role in potential theory, especially in understanding relationships with subharmonic functions, minimum principles, inequalities, and various boundary value problems.

5 Must Know Facts For Your Next Test

1. Superharmonic functions are generally used to describe phenomena like potential theory, where they represent potentials that do not achieve local maxima in their domain.
2. Unlike harmonic functions, superharmonic functions can attain their maxima on the boundary of their domains rather than at interior points.
3. In relation to subharmonic functions, if a function is superharmonic, it implies that the negative of the function is subharmonic.
4. The minimum principle states that a superharmonic function cannot have a minimum value within an open set unless it is constant throughout that set.
5. Superharmonic functions play a critical role in establishing Harnack's inequality, which provides important insights into the behavior of solutions to elliptic partial differential equations.

Review Questions

• How do superharmonic functions relate to subharmonic functions in terms of their properties and implications?
  • Superharmonic functions are essentially the opposites of subharmonic functions. While subharmonic functions are lower semicontinuous and cannot exceed their average values over surrounding spheres, superharmonic functions are upper semicontinuous and do not fall below their average values. If you take the negative of a superharmonic function, it becomes subharmonic. This relationship highlights how these two types of functions interact within potential theory.
• Discuss the implications of the minimum principle for superharmonic functions and how it affects their behavior in a given domain.
  • The minimum principle for superharmonic functions states that they cannot achieve a minimum within an open set unless they are constant throughout that region. This principle implies that superharmonic functions can only attain their minimum values on the boundary of their domains, emphasizing their tendency to 'rise' rather than 'dip' inside a domain. This behavior is critical in applications where one needs to understand potential landscapes or solve related boundary value problems.
• Evaluate how superharmonic functions contribute to understanding Harnack's inequality and its relevance in potential theory.
  • Superharmonic functions are essential for understanding Harnack's inequality, which establishes bounds on positive harmonic functions and relates their values at different points. The properties of superharmonic functions allow us to extend these bounds to cases where we deal with non-negative solutions of elliptic partial differential equations. By leveraging the relationship between superharmonic and harmonic functions, we can gain deeper insights into the regularity and behavior of solutions, making it easier to apply this knowledge in various practical contexts.

Related terms

Subharmonic function: A subharmonic function is a function that is lower semicontinuous and satisfies the mean value property, meaning its value at any point is less than or equal to the average of its values over any surrounding sphere.

Mean value property: The mean value property states that the value of a harmonic function at a point is equal to the average of its values over any sphere centered at that point.

Boundary value problem: A boundary value problem involves finding a function that satisfies a certain differential equation along with specified conditions on the boundary of its domain.