A physical interpretation of a very known elliptical PDE

Question

I am looking for some reference that deals with applications; in real life, of the very known elliptical equation $$ \begin{cases} \begin{aligned} -\Delta u(x) + u(x) &= f(x), x \in \Omega \\ u(x) &= 0, x \in \partial \Omega \end{aligned} \end{cases} $$ For example, in one dimensional case, taking $\Omega = (a,b)$, what would an application of it in real life? Tha is, what does the equation $$ \begin{cases} \begin{aligned} -u''(x) + u(x) &= f(x), x \in (a,b) \\ u(a) = u(b) &= 0 \end{aligned} \end{cases} $$ represents physically?

Answer 1

Generally, for a surface u(x,y), e.g. water/air $\Delta u$ is the surface tension energy density. By integration over a small circle it results into the force on the boundary of a circle. The result is an accelerating net force plus a radial force on the circle trying to flatten local humps, that enlarge the surface area and represent energy concentration areas.

The linear term $u$ is the force back to the 0-level, consider the surface as thee free surface of a system of communicating tubes in a gravitional field.

The inhomogenous term $f$ represents external forces independent of the local amplitude $u$, eg variing gravitational fields or air pressure.

The universality of these three terms for many different phenomena rests in the fact, that $\Delta$ is the euclidean mean operator in a small neighborhood $x \pm dx$ for small deviations from equipartition for time evolutions, eg in dimension 1

$$u_{t+dt}(x) = u_t(x) + dt \ \frac{1}{2} \ (u_t(x+dx)+u_t(x-dx)-2 u_t(x)) = u_t(x) (1-dt\ dx) + \ dt \ \frac{1}{2} ( u_t(x+dx) + u_t(x-dx)) $$