What is "p-laplacian"

In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator, where p is allowed to range over 1 < p < ∞. It is written as

Δu :  = ∇ ⋅ (∣∇u∣∇u).

Where the ∣∇ ⋅ ∣ is defined as

$$\quad |\nabla u|^{p-2} = \left[ \textstyle \left(\frac{\partial u}{\partial x_1}\right)^2 + \cdots + \left(\frac{\partial u}{\partial x_n}\right)^2 \right]^\frac{p-2}{2}$$

In the special case when p = 2, this operator reduces to the usual Laplacian. In general solutions of equations involving the p-Laplacian do not have second order derivatives in classical sense, thus solutions to these equations have to be understood as weak solutions. For example, we say that a function u belonging to the Sobolev space W is a weak solution of

Δu = 0in Ω

if for every test function φ ∈ C we have

∫∣∇u∣∇u ⋅ ∇φ dx = 0

where  ⋅  denotes the standard scalar product.