- #1
thegreenlaser
- 525
- 16
My equation is:
[tex] \left(\mathbf{\nabla}\sigma\right)\cdot\left(\mathbf{\nabla}V\right) + \sigma\nabla^2V = 0[/tex]
If I'm given V(r) on the boundary of some volume, and I know σ(r) inside the volume, is there a unique solution V(r) inside that volume for any arbitrary (well-behaved) function σ(r)?
I suspect the answer is yes, but I've never taken a formal PDE class, so I wanted to double-check.
Edit: just so it's clear, I don't need to know how to solve for V, I just need to know that it's possible to find V in principle.
[tex] \left(\mathbf{\nabla}\sigma\right)\cdot\left(\mathbf{\nabla}V\right) + \sigma\nabla^2V = 0[/tex]
If I'm given V(r) on the boundary of some volume, and I know σ(r) inside the volume, is there a unique solution V(r) inside that volume for any arbitrary (well-behaved) function σ(r)?
I suspect the answer is yes, but I've never taken a formal PDE class, so I wanted to double-check.
Edit: just so it's clear, I don't need to know how to solve for V, I just need to know that it's possible to find V in principle.