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 is allowed to range over. It is written as
Where the is defined as
In the special case when, 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 is a weak solution of
if for every test function we have
where denotes the standard scalar product.The weak solution of the p-Laplace equation with Dirichlet boundary conditions
in a domain is the minimizer of the energy functional
among all functions in the Sobolev space satisfying the boundary conditions in the trace sense. In the particular case and is a ball of radius 1, the weak solution of the problem above can be explicitly computed and is given by
where is a suitable constant depending on the dimension and on only. Observe that for the solution is not twice differentiable in classical sense.
