P-Laplacian Explained

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 < inf ty . It is written as\Delta _{p} u: = \nabla ⋅ (|\nabla u| ^{p-2} \nabla u).

Where the

|\nabla u| ^{p-2} is defined as|\nabla u| ^{p-2} = \left[ style \left( (\partial u)/(\partial x _{1}) \right) ^{2} + … + \left( (\partial u)/(\partial x _{n}) \right) ^{2} \right] ^{(p -2)/(2)}

In the special case when

p =2 , 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 W ^{1,p} (\Omega) is a weak solution of\Delta _{p} u =0 in \Omega

if for every test function

\varphi\in C ^{inf ty} _{0 (\Omega)} we have\int _{\Omega} |\nabla u| ^{p-2} \nabla u ⋅ \nabla\varphidx =0

where

⋅ denotes the standard scalar product.

Energy formulation

The weak solution of the p-Laplace equation with Dirichlet boundary conditions

\begin{cases} - \Delta _{p} u = f& in \Omega\\ u = g & on \partial\Omega \end{cases}

in a domain

\Omega\subset R ^{N} is the minimizer of the energy functionalJ(u) = (1)/(p) \int _{\Omega} |\nabla u| ^{p} dx - \int _{\Omega} fudx W ^{1,p} (\Omega) satisfying the boundary conditions in the trace sense.^[1] In the particular case f =1 , g =0 and \Omega is a ball of radius 1, the weak solution of the problem above can be explicitly computed and is given byu(x) = C \left( 1- |x| ^{(p)/(p -1)} \right)

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.

See also

• Infinity Laplacian

Sources

• Evans . Lawrence C. . Lawrence C. Evans . A New Proof of Local C ^{1,\alpha} Regularity for Solutions of Certain Degenerate Elliptic P.D.E. . . 45 . 356–373 . 1982. 672713 . 10.1016/0022-0396(82)90033-x. free .
• Lewis . John L. . Capacitary functions in convex rings . . 66 . 201–224 . 1977. 3 . 0477094 . 10.1007/bf00250671. 1977ArRMA..66..201L . 120469946 .

Further reading

• .
• Uhlenbeck . K. . Karen Uhlenbeck . Regularity for a class of non-linear elliptic systems . . 138 . 219–240 . 1977. 0474389 . 10.1007/bf02392316. free .
• Notes on the p-Laplace equation by Peter Lindqvist
• Juan Manfredi, Strong comparison Principle for p-harmonic functions

Notes and References

1. Evans, pp 356.