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 $p$ is allowed to range over $1<p<\mathrm{\infty }$. It is written as

${\mathrm{\Delta }}_{p}u:=\mathrm{d}\mathrm{i}\mathrm{v}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u).$

Where the $|\mathrm{\nabla }u{|}^{p-2}$ is defined as

$|\mathrm{\nabla }u{|}^{p-2}={\left[{\left(\frac{\partial u}{\partial x_1}\right)}^2+\cdots +{\left(\frac{\partial u}{\partial x_n}\right)}^2\right]}^{\frac{p-2}{2}}$

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 $W^{1,p}(\mathrm{\Omega })$ is a weak solution of

${\mathrm{\Delta }}_{p}u=0\text{ in }\mathrm{\Omega }$

if for every test function $\phi \in C_0^{\mathrm{\infty }}(\mathrm{\Omega })$ we have

$\underset{\mathrm{\Omega }}{\int }|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\cdot \mathrm{\nabla }\phi dx=0$

where $\cdot$ denotes the standard scalar product.

Energy formulation

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

$\{\begin{array}{cc}-{\mathrm{\Delta }}_{p}u=f& \text{ in }\mathrm{\Omega }\\ u=g& \text{ on }\partial \mathrm{\Omega }\end{array}$

in an open bounded set $\mathrm{\Omega }\subseteq {ℝ}^N$ is the minimizer of the energy functional

$J(u)=\frac{1}{p}\underset{\mathrm{\Omega }}{\int }|\mathrm{\nabla }u{|}^{p}dx-\underset{\mathrm{\Omega }}{\int }fudx$

among all functions in the Sobolev space $W^{1,p}(\mathrm{\Omega })$ satisfying the boundary conditions in the sense that $u-g\in W_0^{1,p}(\mathrm{\Omega })$ (when $\mathrm{\Omega }$ has a smooth boundary, this is equivalent to require that functions coincide with the boundary datum in trace sense). In the particular case $f=1,g=0$ and $\mathrm{\Omega }$ is a ball of radius 1, the weak solution of the problem above can be explicitly computed and is given by

$u(x)=C(1-|x{|}^{\frac{p}{p-1}})$

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.

See also

• Infinity Laplacian

Sources

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Further reading

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• Notes on the p-Laplace equation by Peter Lindqvist
• Juan Manfredi, Strong comparison Principle for p-harmonic functions

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