Physical interpretation of $-\Delta u = \lambda u$ on $\Omega$ with boundary conditions $u|_{\partial \Omega} = 0$

Question

Let $\Omega$ be a (regular) domain in $\mathbb{R}^d$, let $\lambda \in \mathbb{R}$ and let $u \colon \Omega \to \mathbb{R}$ be a non-null (regular) function such that $$-\Delta u = \lambda u \quad \text{ on } \Omega \;, \qquad u|_{\partial \Omega} = 0$$ i.e., $u$ is an eigenfunction relative to the eigenvalue $\lambda$ for the Laplace operator with vanishing boundary Dirichlet conditions.

To better grasp the intuition behind this equation, I'm wondering if we can name some scalar field $u : \Omega \to \mathbb{R}$ coming from physics (something like temperature, or pression, or potential, or energy, or whatever, with $d = 1,2,3$ or maybe even $d =4$) satisfying the above conditions for some $\lambda \neq 0$.

What I'm looking for is something in the spirit of the following examples:

1. Solutions to $- \Delta u = 0$ model the potential of the electrostatic field in a zone without charges
2. Solutions to $- \Delta u = \lambda \cdot \partial_tu$ model the evolution of the temperature in a room
3. Solutions to $-\Delta u = \lambda \cdot \partial_t^2u$ model the evolution of electromagnetic waves

but I've no clear idea of what the equation $-\Delta u = \lambda u$ is modeling...

References and pointers are welcome.

Answer 1

Consider the wave equation in one dimension $$ \partial_t^2 u(t,x) = c^2\partial_x^2 u(t,x). $$ If we assume the wave to be time harmonic $u(t,x)=e^{i\omega t}u(x)$ for some $\omega$, then $u(x)$ is solution to the Helmholtz equation $$ -\omega^2 u(x) = c^2\partial_x^2 u(x), $$ that is, the question you are interested in.

The boundary condition just states that you force your wave to be zero on the boundary.

Answer 2

What you are looking at is called the Helmholtz equation. To get an intuition I recommend to start with the simplest case d=1. This gives us the differential equation:

$- \Delta u = \lambda u$ =

$- \frac{\partial^2 u(x)}{\partial x^2} = \lambda u(x)$ =

$- u''(x) = \lambda u(x)$

for some variable x (assuming in $R$). The general solution to this equation is the $Asin(x\sqrt\lambda)$+$Bcos(x\sqrt\lambda)$, which we interpret as a wave (for some constants A,B and $ \lambda \in R$). Increasing the dimension leads to looking at a problem of similar nature, where the solution (which can be obtained for simple geometries by the separation of variables) represents wave shapes of more complex nature as the wave can propagate in more ways than in one dimension.

Answer 3

There are very nice interpretations of these eigenfuntions in the book Numerical computing with MATLAB by Cleve Moler, in the chapter on PDEs:

https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/pdes.pdf

The relevant section is 11.5: The L-shaped membrane. Incidentally, the famous MathWorks logo (below) is the graph of one such eigenfunction, with $\Omega$ being a domain shaped like a letter $L$; hence the title of the section.