MHB Finding values for a harmonic function

  • Thread starter Thread starter Stephen88
  • Start date Start date
  • Tags Tags
    Function Harmonic
Click For Summary
The discussion focuses on finding values for a harmonic function defined in a closed disc with specific boundary conditions. The value at the center of the disc can be determined using the mean value property for harmonic functions, which states that the value at any point is the average of the values on the boundary. For the maximum and minimum values on the closed disc, the maximum and minimum principle for harmonic functions indicates that these extrema occur on the boundary. The boundary values are given by the function sin(θ) + cos(θ), which can be analyzed to find the required maximum and minimum. Overall, the problem emphasizes the application of harmonic function properties to derive values without explicitly solving for the function itself.
Stephen88
Messages
60
Reaction score
0
A function u is harmonic in a domain containing the closed disc x^2+y^2 ≤ 1. Its values on the boundary
are given in terms of the polar angle # by sin# + cos#. Without finding u find
a. its value at the centre of the disc
b. its maximum and minimum values on the closed disc.
The disc is probably defined as D(x,y),r) where r=x^2+y^2 ≤ 1 and I think that the max and minimum can be find on the boundary of the set.But I think that u must also be continuous for that to happen.
Can someone give me a detailed explanation or an example on how to solve this problem?
 
Physics news on Phys.org
What is the best way to solve a and b?
 
For a use the mean value property for harmonic functions. For b use the maximum and minimum principle.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
View full post »

Similar threads

MHB Global Extrema of a Trigonometric Function
  • · Replies 16 ·
Replies
16
Views
2K
Graduate Proving that this integral is divergent
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Complex Analysis Harmonic functions
  • · Replies 3 ·
Replies
3
Views
2K
MHB Dirichlet problem for the laplacian in the strip
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Compact support functions and law of a random variable
  • · Replies 11 ·
Replies
11
Views
2K
Harmonic Functions: Laplace's Equations & Analytic Functions
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Are slices of measurable functions measurable?
  • · Replies 3 ·
Replies
3
Views
3K
MHB Determining the Maximum and Minimum of a Discontinuous Function $f(x,y)$
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How do I separate the real and imaginary parts of an infinite product?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Regarding fibrations between smooth manifolds
  • · Replies 1 ·
Replies
1
Views
2K