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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.[1] 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.

Energy formulation[edit]

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.[1] 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.


  1. ^ a b Evans, pp 356.


  • Evans, Lawrence C. (1982). "A New Proof of Local Regularity for Solutions of Certain Degenerate Elliptic P.D.E.". Journal of Differential Equations. 45: 356–373. doi:10.1016/0022-0396(82)90033-x. MR 0672713.
  • Lewis, John L. (1977). "Capacitary functions in convex rings". Archive for Rational Mechanics and Analysis. 66: 201–224. doi:10.1007/bf00250671. MR 0477094.

Further reading[edit]