P-Laplacian Walk

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 $1 < p < \infty$. It is written as

:$\Delta_p u:=\nabla \cdot (, \nabla u, ^ \nabla u).$

Where the $, \nabla u, ^$ is defined as

:\quad , \nabla u, ^ = \left \textstyle \left(\frac\right)^2
+ \cdots + \left(\frac\right)^2
\right\frac

In the special case when $p=2$, this operator reduces to the usual Laplacian.Evans, pp 356. 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 $W^(\Omega)$ is a weak solution of

:$\Delta_p u=0 \mbox \Omega$

if for every test function $\varphi\in C^\infty_0(\Omega)$ we have

:$\int_\Omega , \nabla u, ^ \nabla u\cdot \nabla\varphi\,dx=0$

where $\cdot$ denotes the standard scalar product.

Energy formulation

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

:$\begin -\Delta_p u = f& \mbox\Omega\\ u=g & \mbox\partial\Omega \end$

in a domain $\Omega\subset\mathbb^N$ is the minimizer of the energy functional

:$J(u) = \frac\,\int_\Omega , \nabla u, ^p \,dx-\int_\Omega f\,u\,dx$

among all functions in the Sobolev space $W^(\Omega)$ satisfying the boundary conditions in the trace sense. 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 by

:$u(x)=C\, \left(1-, x, ^\frac\right)$

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.

Notes

Sources

*
*

Further reading

*.
*
Notes on the p-Laplace equation
by Peter Lindqvist

Juan Manfredi, Strong comparison Principle for p-harmonic functions

Elliptic partial differential equations

{{mathanalysis-stub