What is the meaning of radial solution for Laplace's equation?

Question

In Evan's book of Partial Differential Equation, for Laplace's equation it is noted that: "In looking for explicit solutions, it is often wise to restrict attention to classes of functions with certain symmetry properties. Since Laplace's equation is invariant under rotations, it consequently seems advisable to search for radial solutions, that is, functions of $r=|x| = (x_1^2+\dots+x_n^2)^{\frac{1}{2}}$."

This radial solution leads to the fundamental solution of Laplace's equation. I know the mathematical definition of symmetry and radial solution which I found in the Internet where the independent variable is a spatial variable varying over a domain with radial symmetry such as a ball centered at the origin.

My questions are:

• What is the relation of radial solution with the symmetry property?
• Is there any other solution (instead of radial solution), to derive the fundamental solution? Is this also need to be related to symmetry property?
• Should we consider radial solutions for any symmetric differential operator to obtain the fundamental solution?

Thanks in advance.

Answer 1

What is the relation of radial solution with the symmetry property?

"Radial" is the short form of "radially symmetric". So you are asking about the relation of symmetric solutions with the symmetry property. The relation is that symmetric things have the property of symmetry. It's a tautology.

Is there any other solution (instead of radial solution), to derive the fundamental solution?

A fundamental solution is not necessarily symmetric. One can use, for example, $u(x,y) = \frac{1}{2\pi}\log\sqrt{x^2+y^2} +3x - 5y$ as a fundamental solution of the Laplace equation in two dimensions. It's not radially symmetric. It's a fundamental solution because it satisfies $\Delta u = \delta_0$, the delta-function.

Should we consider radial solutions for any symmetric differential operator to obtain the fundamental solution?

Yes. The fundamental solution for $L$ is a function $u$ such that $Lu$ is the delta-function at $0$. The delta-function is a radially symmetric distribution. So it is natural to look for fundamental solutions that are symmetric. It simplifies the search.

Answer 2

A radial solution in solving partial differential equations isolates the part of the general solution dependent solely on the radial coordinate "r," distinguishing it from the angular components. This separation, grounded in the assumption of variable separation, doesn't inherently imply symmetry or invariance along the radial direction. While the radial part captures the radial dependence, it may lack symmetry. Introducing symmetry considerations, particularly in the angular part, imposes additional constraints on separation constants, enhancing our understanding of the solution's behavior in specific contexts.