I am studying the numerical aspects of fourth-order elliptic problems now, and I came across the plate problem:
Let $\Omega\subset\mathbb{R}^n$ bounded domain with Lipschitz-Boundary. Find $u$ s.t. $\Delta^2 u = f$, on $\Omega$ and $\partial_\nu u = 0$ on the boundary $\Gamma=\partial\Omega$. $\partial_\nu = \nabla u \cdot \nu$ is the derivative in direction of the outer normal
Finding a weak formulation, construction of the FE Spaces, etc. seem very clear to me. My Problem ist just, that i cannot imagine what's the underlying problem described in those equations.
I also tried to look it up (e.g. in Ciarlet), but it is still not clear to me at all what the problem is. The name suggests, that it is about a plate, and the only thing I can imagine doing with a plate is apply force to the boundary and see how it deforms.
This would mean that $u$ is the displacement?
Is it reasonable to just consider it without time-dependency? What does that mean (equilibrium)?
It would be very nice, if someone could explain the problem a bit.
Related to this, there's also a clamped plate problem, and also in this case I don't have a clue what's described here.
The underlying problem is that the plate is subjected to distributed load with density $f$, and is fixed along the boundary in such a way that the boundary can neither mode nor rotate (clamping condition). The function $u$ represents vertical displacement of the plate when it has reached an equilibrium. Since an equilibrium is being studied, no time is involved in the equation.
The PDE can be derived from energy minimization. The total energy of the system, to be minimized, consists of the bending energy $\int |\Delta u|^2$ (this is an idealized energy functional), and the potential energy $-\int uf$. Minimizing the sum $\int |\Delta u|^2 - uf$ leads to the Euler-Lagrange equation $\Delta^2 u-f=0$ (possibly with some factor of $2$ or $4$ that I neglected).