**Motivation**


I encountered the following partial differential equation (PDE) in a [mathematical paper](http://www.sciencedirect.com/science/article/pii/S0362546X14000959)


$$\begin{array}{}
u\_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)
\\\qquad\quad-\Delta u\_{t}+\int\_{0}^{t}g(t-s)\Delta u(x,s) ds=f(x,u,u\_{t}) & \text{in} & \partial \Omega \times (0,T) \\
u=\frac{\partial u}{\partial n}=0 & \text{on} & \partial \Omega \times [0,T) \\
u(x,0)=u\_0(x), \qquad u\_{t}(x,0)=v\_0(x) & \text{in} & \partial \Omega
\end{array} $$


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](http://www.math.kit.edu/iana3/~schnaubelt/media/dampedplate.pdf) that if we have **structural damping** then the term $\Delta u\_t$ shows up in $(1)$. Also, [an article in wikipedia](https://en.wikipedia.org/wiki/Viscoelasticity#Types_of_viscoelasticity) 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.




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**Question**


Can you please shed some light on the **physical**, **mathematical** or **historical** background of the p-Laplacian term? Where does it come from?



Backgrounds of the p-Laplacian Operator
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.




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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.




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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.




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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.




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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?


