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.


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

  • 2
    $\begingroup$ As you have observed, each term in the equation by itself has a physical motivation, but the combination is not so easily motivated. The true motivation is probably that this combination happens to be something for which the authors succeeded in getting estimates. It is the motivation behind many papers. $\endgroup$ Jan 11, 2016 at 9:37

3 Answers 3


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?


The limits $p\to 1$ and $p\to \infty$ have geometrical significance: The 1-Laplace operator $$\Delta_1 u=\nabla \cdot \left ( \frac{\nabla u}{|\nabla u|} \right )$$ measures the mean curvature of the level set in each point, and the (homogenized) $\infty$-Laplace operator $$\Delta_\infty u=D^2 u \frac{\nabla u}{|\nabla u|}\cdot \frac{\nabla u}{|\nabla u|}$$ is the second derivative in the direction of steepest ascent. One can then view the (homogenized) $p$-Laplace operator as a weighted sum of these two extremal operators $$\frac{1}{p}|\nabla u|^{2-p} \Delta_p u=\frac{1}{p} |\nabla u|\Delta_1 u+\frac{p-1}{p} \Delta_\infty u$$ For $p=2$ the weights are equal, for $p<2$ the curvature part weights more, and for $p>2$ the second derivative part weights more.


In the following paper you can find the historical process of derivation of the p-Laplace operator https://ejde.math.txstate.edu/Volumes/2018/16/benedikt.pdf


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