- #1
bdforbes
- 152
- 0
I can't figure out how to use the Green's function approach rigorously, i.e., taking into account the fact that the Dirac Delta function is not a function on the reals.
Suppose we have Laplace's Equation:
[tex]\nabla^2 \phi(\vec{x})=f(\vec{x})[/tex]
The solution, for "well-behaved" [itex]f(\vec{x})[/itex] is
[tex]\phi(\vec{x})=\frac{-1}{4\pi}\int \frac{f(\vec{x}')}{\left|\vec{x}-\vec{x}'\right|}d^3\vec{x}'[/tex]
It is my understanding that this integral is well-defined as both a Riemann-Darboux and Lebesgue integral. If we treat it as a Lebesgue integral, I believe the limiting operations can be exchanged, i.e., we can apply the Laplacian to the integrand:
[tex]\nabla^2 \phi(\vec{x})=\frac{-1}{4\pi}\int\nabla^2\left(\frac{1}{\left|\vec{x}-\vec{x}'\right|}\right)f(\vec{x}')d^3\vec{x}'[/tex]
But now it looks like this Lebesgue integral is NOT well-defined! How do we deal with [itex]\nabla^2(1/|x-x'|)[/itex] at the singular point?
If we naively apply the divergence theorem, we can arrive at the desired result, but that is not good enough for me.
How can we do this rigorously? Is there a way to use the Dirac measure, or the Dirac Delta as a linear functional?
Suppose we have Laplace's Equation:
[tex]\nabla^2 \phi(\vec{x})=f(\vec{x})[/tex]
The solution, for "well-behaved" [itex]f(\vec{x})[/itex] is
[tex]\phi(\vec{x})=\frac{-1}{4\pi}\int \frac{f(\vec{x}')}{\left|\vec{x}-\vec{x}'\right|}d^3\vec{x}'[/tex]
It is my understanding that this integral is well-defined as both a Riemann-Darboux and Lebesgue integral. If we treat it as a Lebesgue integral, I believe the limiting operations can be exchanged, i.e., we can apply the Laplacian to the integrand:
[tex]\nabla^2 \phi(\vec{x})=\frac{-1}{4\pi}\int\nabla^2\left(\frac{1}{\left|\vec{x}-\vec{x}'\right|}\right)f(\vec{x}')d^3\vec{x}'[/tex]
But now it looks like this Lebesgue integral is NOT well-defined! How do we deal with [itex]\nabla^2(1/|x-x'|)[/itex] at the singular point?
If we naively apply the divergence theorem, we can arrive at the desired result, but that is not good enough for me.
How can we do this rigorously? Is there a way to use the Dirac measure, or the Dirac Delta as a linear functional?