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I'm trying to find solutions for the Poisson equation under Neumann conditions, and have a couple of questions. More specifically, I'm interested in the gradient of the function $\phi(x)$ in a space $\Omega \subset \mathbb{R}^d$. (note that I'm only interested in the gradient. For my problem I do not care about $\phi(x)$ at all. I know two things about $\phi(x)$. First, I know the Laplacian on the entire set $\Omega$: $$ \nabla^2 \phi(x)=f(x)\quad \forall \quad x \in \Omega $$ Second, the following boundary condition: $$ \nabla \phi(x)n=0 \quad \forall \quad x \in \partial \Omega $$ where n is the outward unit normal to $\Omega$. As I understand it, the solution for $\phi(x)$ is given by: $$ \phi(x_0)=\int_\Omega f(x) G(x,x_0) dx + boundary terms+arbitrary constant $$ And my object of interest, the gradient of $\phi$ is given by: $$ \nabla_{x_0} \phi(x_0)=\int_\Omega f(x) \nabla_{x_0} G(x,x_0) dx +\nabla_{x_0} boundary terms $$ where $G$ is the Green function of my problem.

I have a couple of questions:

  1. Does anybody know a text that works out this problem under Neumann conditions? I have seen many treatises where they look at Dirchilet conditions, but none with Neumann. I would particularly be interested in how to define the Green function exactly.

  2. Are my boundary terms zero (because of the rather simple boundary condition) in the problem?

  3. I'm trying to get a feel for the Green's function in different spaces. Is defining the Green's function in this problem somehow similar to determining the appropriate bounds for integration? For example, suppose that the problem occurs in only 1 dimension. In that case, the gradient of the Green's function, $\nabla_{x_0} G(x,x_0)$ should be a stepwise function that takes value 1 for all $x$ smaller than $x_0$ and value 0 thereafter right? To me this seems to be the only way to retrieve the standard solution for a one-dimensional problem.

  4. Should it not be easier to retrieve the gradient of the Green's function (which I'm interested in), rather than the Green's function itself? Is there any text that treats this issue?

Many thanks for any help you can offer.

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  • $\begingroup$ It may be helpful if you try to focus on a specific question rather than a bunch of related ones. $\endgroup$ – jvriesem Jul 21 '15 at 17:44
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1. Does anybody know a text that works out this problem under Neumann conditions?

I do not. However, I've had a similar problem and think the following may be useful to you:

2. Are my boundary terms zero (because of the rather simple boundary condition) in the problem?

The boundary terms ($\phi(x_{min})$ and $\phi(x_{max})$ in 1D) aren't necessarily zero. Since the PDE itself and the boundary conditions all involve only derivatives of $\phi(x)$, there is nothing tying the solution down to any particular value of $\phi(x)$. That means the solution will be unique only up to an additive constant. As a result, one solution will see at least one of the endpoints having a value of zero, but there will an infinite number of solutions which won't have zero on the boundary. For more information, see Uniqueness of solutions to the Laplace and Poisson equations.

3. Is defining the Green's function in this problem somehow similar to determining the appropriate bounds for integration?

I'm not sure about the particulars for this, unfortunately. I do know that using Green's functions are intimately connected to the particular geometry and the PDE in a given problem. However, as can be seen from the Green's function for electrostatics ($G(\vec{x};\vec{x}') = \tfrac{1}{\left| \vec{x} - \vec{x}' \right|}$), neither the geometry nor the boundary conditions can be seen in the definition of the Green's function. This document has several examples and may be useful to you.

4. Should it not be easier to retrieve the gradient of the Green's function (which I'm interested in), rather than the Green's function itself? Is there any text that treats this issue?

I have no idea! Sorry! The links above might (hopefully!?) help.

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