Motivation
I encountered the following partial differential equation (PDE) in a mathematical paper
$$\begin{array}{}
u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)
\\\qquad\quad-\Delta u_{t}+\int_{0}^{t}g(t-s)\Delta u(x,s) ds=f(x,u,u_{t}) & \text{in} & \partial \Omega \times (0,T) \\
u=\frac{\partial u}{\partial n}=0 & \text{on} & \partial \Omega \times [0,T) \\
u(x,0)=u_0(x), \qquad u_{t}(x,0)=v_0(x) & \text{in} & \partial \Omega
\end{array} $$
where $\Omega \subset \mathbb{R}^n$ is a bounded domain with Lipschitz-continuous boundary $\partial \Omega$. Also, $g \ge 0$ is called memory kernel that decays with a general rate and $f(x,u,u_t)$ is some nonlinear function. $p \ge 2$ is a real contant. The differential operators $\Delta$, $\Delta^2$ and $\nabla$ are the Laplacian, the Biharmonic and gradient operators, respectively.
We are interested in the case $n=3$ which has a physical meaning. Now, let us go into some physical insights.
Plates are initially flat structural members bounded by two parallel planes, called faces, and a cylindrical surface, called an edge or boundary. The generators of the cylindrical surface are perpendicular to the plane faces. The deck of a ship is an example of a plate. This PDE is describing the lateral displacement of a plate made of a homogeneous isotropic nonelinear viscoelastic material.
The function $u(x,t)$ is the lateral displacement of the plate at position $x$ and time $t$. I know that in classical linear elasticity, the following equation
$$u_{tt}+\Delta^2u=f(x) \tag{1}$$
describes the lateral displacement of a plate made of a homogeneous isotropic elastic material where $f(x)$ is an external force applied to the plate and $u_{tt}$ describes the inertia or acceleration term. It is also known as the equation for vibration of plates. I found from this paper that if we have structural damping then the term $\Delta u_t$ shows up in $(1)$. Also, an article in wikipedia revealed that the integral term $\int_{0}^{t}g(t-s)\Delta u(x,s) ds$ can show up in $(1)$ when viscoelasticity comes in. What remains unknown is
$$\Delta_p u \equiv \nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)$$
which is called the p-Laplacian operator. For $p=2$, it is the usual Laplacian operator $\Delta$. I really cannot find any background of this operator.
Question
Can you please shed some light on the physical, mathematical or historical background of the p-Laplacian term? Where does it come from?
Best Answer
I'm writing about the scalar-valued equation, so the solution is $u \colon \Omega \to \mathbb{R}$ and $\Omega \subset \mathbb{R}^d$, $d \geq 1$. Maybe this is of some use.
Basic lecture notes on $p$-Laplace equation, mathematical aspects: http://www.math.ntnu.no/~lqvist/p-laplace.pdf . See also the bibliography.
The $p$-Laplace equation is a prototype of nonlinear (or quasilinear) elliptic PDE and has many properties that resemble those of the 2-Laplace equation.
Variational justification: The solutions of 2-Laplace equation minimize the energy $$\int_\Omega |\nabla u|^2 \text{d} x$$ in the space $H^1 (\Omega) = W^{1,2} (\Omega)$ with fixed Dirichlet boundary conditions.
Solutions of the $p$-Laplace equation minimize the energy $$\int_\Omega |\nabla u|^p \text{d} x$$ in the space $W^{1,p} (\Omega)$ with fixed Dirichlet boundary conditions.
One possible physical interpretation is conductivity of electricity. In your situation there should also be some power-law behaviour.
Recall the Ohm's law, which states that current flux $j$ is proportional to differences in electric potential $\nabla u$ (I assume constant conductivity); $$-j = \nabla u.$$ By Kirchhoff's law you have $\nabla \cdot j = 0$ when there are no sources or sinks of electricity. Combine these and you have the Laplace equation $$-\Delta u = 0.$$
The Ohm's law is only an approximation; in reality, you can have complicated non-linear relations there. One possible relation is of power-law type, where $$-j = |\nabla u|^{p-2}\nabla u,$$ which leads to the $p$-Laplace equation. This power law relation has been observed in some materials near the temperatures where the material becomes superconductive; there $p$ is a function of temperature.
On history: I have a faint memory of someone saying that the origin of $p$-Laplace equation is in (non-linear) fluid dynamics. I have not checked this out. I guess Ladyzhenskaja would be a likely author. Perhaps investigate there?