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
Δ p u := ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) . Where the | ∇ ⋅ | p − 2 is defined as
| ∇ u | p − 2 = [ ( ∂ u ∂ x 1 ) 2 + ⋯ + ( ∂ u ∂ x n ) 2 ] 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 1 , p ( Ω ) is a weak solution of
Δ p u = 0 in Ω if for every test function φ ∈ C 0 ∞ ( Ω ) we have
∫ Ω | ∇ u | p − 2 ∇ u ⋅ ∇ φ d x = 0 where ⋅ denotes the standard scalar product.
The weak solution of the p-Laplace equation with Dirichlet boundary conditions
{ − Δ p u = f in Ω u = g on ∂ Ω in a domain Ω ⊂ R N is the minimizer of the energy functional
J ( u ) = 1 p ∫ Ω | ∇ u | p d x − ∫ Ω f u d x among all functions in the Sobolev space W 1 , p ( Ω ) satisfying the boundary conditions in the trace sense. In the particular case f = 1 , g = 0 and Ω is a ball of radius 1, the weak solution of the problem above can be explicitly computed and is given by
u ( x ) = C ( 1 − | x | p p − 1 ) where C is a suitable constant depending on the dimension N and on p only. Observe that for p > 2 the solution is not twice differentiable in classical sense.
