Normal derivative at boundary Laplace's equation half plane

In summary, the Poisson integral formula provides a solution to a holomorphic function u(x,y) defined in the upper half-plane, given a boundary value of f(x). The formula involves an integral and can be used to find the partial derivative of u(x,y) with respect to y at y=0. The process for taking this limit may require further research or references for a more accurate answer.
  • #1
nickthequick
53
0
Hi,

Given a holomorphic function [itex] u(x,y) [/itex] defined in the half plane ([itex] x\in (-\infty,\infty), y\in (-\infty,0)[/itex]), with boundary value [itex] u(x,0) = f(x) [/itex], the solution to this equation (known as the Poisson integral formula) is

[tex] u(x,y) = \int_{-\infty}^{\infty} \frac{y\ f(t) }{(t-x)^2 +y^2} \ dt [/tex].


I have two questions. First, how does one prove

[itex] u(x,0) = f(x) [/itex],

and related, but what I'm more interested in, how do I find

[itex] \left.\frac{\partial u(x,y) }{\partial y}\right|_{y=0} [/itex]


The subtleties on how take these limits (namely [itex] y\to 0; \ y<0[/itex]) appropriately is very much beyond me. Any suggestions/references on how to answer this would be very appreciated.

Thanks!

Nick
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
I'm not entirely sure about your first question. You defined the upper half-plane problem to have the BV of f(x). I don't think you would need to prove that, because that is just a definition in the problem. Correct me if I'm wrong.
 

1. What is the definition of a normal derivative at the boundary in Laplace's equation on a half plane?

The normal derivative at the boundary in Laplace's equation on a half plane is the derivative of the solution to the equation with respect to the direction perpendicular to the boundary. It represents the rate of change of the solution along the normal direction at the boundary.

2. Why is the normal derivative at the boundary important in Laplace's equation on a half plane?

The normal derivative at the boundary is important because it allows us to determine the behavior of the solution to Laplace's equation on a half plane near the boundary. It helps us understand how the solution changes in the normal direction at the boundary, which is essential for solving boundary value problems.

3. How is the normal derivative at the boundary calculated for Laplace's equation on a half plane?

The normal derivative at the boundary can be calculated using the formula:

$\frac{\partial u}{\partial n} = \nabla u \cdot \vec{n}$

where $\frac{\partial u}{\partial n}$ is the normal derivative, $\nabla u$ is the gradient of the solution, and $\vec{n}$ is the normal vector at the boundary.

4. Can the normal derivative at the boundary be negative in Laplace's equation on a half plane?

Yes, the normal derivative at the boundary can be negative in Laplace's equation on a half plane. This means that the solution decreases in the normal direction at the boundary. However, the normal derivative is typically defined as positive if it points outward from the boundary.

5. How can the normal derivative at the boundary be used to solve boundary value problems in Laplace's equation on a half plane?

The normal derivative at the boundary can be used to solve boundary value problems by applying boundary conditions. For example, if the boundary condition specifies the normal derivative at the boundary, it can be used in conjunction with the formula to determine the solution at that point. Additionally, the normal derivative can be used to construct the Green's function, which is a fundamental tool for solving boundary value problems in Laplace's equation.

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