Find all solutions u=f(r) of the three-dimensional Laplace...

Find all solutions $u=f(r)$ of the three-dimensional Laplace equation $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}=0$ that depend only on the radial coordinate $r=\sqrt{x^2+y^2+z^2}$. Do these solutions form a vector space? If so, what is its dimension?

Find all solutions $u=f(r)$ of the three-dimensional Laplace equation $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}=0$ that depend only on the radial coordinate $r=\sqrt{x^2+y^2+z^2}$.
Do these solutions form a vector space? If so, what is its dimension?

Key Concepts

Laplace's Equation

Laplace's equation is a second-order partial differential equation commonly expressed as ?²u = 0. It arises in various fields such as electrostatics, gravitation, and fluid dynamics, where it describes steady-state heat distribution, potential fields, and more. Understanding the nature of this equation is crucial for identifying its solutions under different conditions.

Radial Symmetry

Radial symmetry refers to a scenario where a function depends only on the distance from a central point rather than on individual coordinates. In the context of the Laplace equation, assuming u = f(r) reduces the three-dimensional problem into one that involves only the radial variable, greatly simplifying the analysis and solution process.

Reduction to an Ordinary Differential Equation

When a function exhibits radial symmetry, the three-dimensional Laplace equation reduces to an ordinary differential equation (ODE) in the radial coordinate. This reduction involves transforming the Laplacian in Cartesian coordinates into a form that depends solely on the radial derivative, which is key to simplifying and solving the problem.

General Solution of the Radial ODE

After reduction, the resulting ODE, typically of the form f''(r) + (2/r) f'(r) = 0, can be solved to yield the general solution involving two arbitrary constants. This highlights the process of finding explicit solutions that depend only on r, and illustrates how symmetry can simplify partial differential equations.

Vector Space of Solutions

The set of solutions of a linear differential equation forms a vector space because the sum of any two solutions is also a solution, and any scalar multiple of a solution is also a solution. In this case, the space of radially symmetric solutions of the Laplace equation is two-dimensional, as evidenced by the two arbitrary constants in the general solution.

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