Discussion Overview
The discussion centers on finding an expression for the Green's function for Poisson's equation that vanishes on an ellipse in two dimensions. Participants explore potential methods and references related to the construction of this Green's function, particularly in relation to elliptical coordinates and existing solutions for circular geometries.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant inquires about the existence of a Green's function for Poisson's equation that vanishes on an ellipse, noting the known solution for a circle.
- Another participant suggests that Laplace's equation on an ellipse is separable in elliptical coordinates and can be expressed using Mathieu functions, which may aid in constructing the Green's function.
- A different participant expresses concern that solving Laplace's equation may not be applicable since the Laplacian of the Green's function equals the Dirac delta, not zero, and questions whether the same technique is valid.
- One participant argues that the Green's function equation resembles Laplace's equation except at a singular point, implying that solutions to Laplace's equation could be used to derive the Green's function, while also acknowledging the potential for various methods involving elliptical coordinates and Mathieu functions.
- Another participant expresses intent to search for more information based on the suggestions provided.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the best approach to derive the Green's function for the elliptical case, with multiple competing views on the applicability of techniques and the relevance of existing solutions.
Contextual Notes
Participants mention the need for references and the complexity of the algebra involved in constructing the Green's function, indicating that the discussion may depend on specific mathematical techniques and definitions related to elliptical coordinates and Mathieu functions.