Poisson's Formula for the half space

In summary, we discussed the map that takes a point in the set into the reflected point in the set . We also introduced the fundamental solution and the green's function for the problem in with boundary condition on . We then showed the outward-pointing unit normal vector field on the boundary and how to compute the derivative of in the direction of this vector field
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Homework Statement
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Relevant Equations
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Hi all, thanks in advance for any constructive feedback. :bow:

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Definition:
If then the map takes the point into given by

We take the reflected point and the fundamental solution
then

and hence is the green's function for

The outward pointing unit normal vector field on the boundary is given by a function that assigns a unit vector to every point on .

The differential operator

The derivative of in the direction of the vector field is given by


Have I made mistakes so far, and can I continue working?
 
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update with "progress"

Definition:
If , then the map takes every point into

To solve the problem

We consider the reflected point and the fundamental solution . The following problem

gives the corrector function

and hence the green's function is
The outward-pointing unit normal vector field on the set is given by a function that assigns the unit vector to every point on the set . Let us take the differential operator
and compute the derivative of in the direction on the set in the flat Euclidean connection
 

FAQ: Poisson's Formula for the half space

What is Poisson's Formula for the half space?

Poisson's Formula for the half space is a mathematical formula that relates the behavior of a harmonic function on the boundary of a half space to its behavior inside the half space. It is a fundamental tool in the study of partial differential equations.

How is Poisson's Formula derived?

Poisson's Formula is derived using the method of images, which involves creating a "mirror image" of the original boundary function and using it to solve the problem. This technique is based on the principle of superposition, where the solution to a problem is the sum of individual solutions to simpler problems.

What are the applications of Poisson's Formula for the half space?

Poisson's Formula has a wide range of applications in physics, engineering, and other fields. It is commonly used to solve problems in electrostatics, heat conduction, fluid mechanics, and potential theory. It is also a useful tool in solving boundary value problems for partial differential equations.

Can Poisson's Formula be extended to higher dimensions?

Yes, Poisson's Formula can be extended to higher dimensions, such as the full space or a general domain. However, the method of images is not applicable in higher dimensions, and other techniques must be used to derive the formula.

Are there any limitations to using Poisson's Formula for the half space?

While Poisson's Formula is a powerful tool for solving boundary value problems, it does have some limitations. It can only be applied to linear problems, and the boundary conditions must be specified on a flat boundary. It also assumes that the solution is continuous and differentiable inside the half space.

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