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 {\displaystyle p} is allowed to range over 1 < p < ∞ {\displaystyle 1<p<\infty } . It is written as
Δ p u := d i v ( | ∇ u | p − 2 ∇ u ) . {\displaystyle \Delta _{p}u:=\mathrm {div} (|\nabla u|^{p-2}\nabla u).}
Where the | ∇ u | p − 2 {\displaystyle |\nabla u|^{p-2}} is defined as
| ∇ u | p − 2 = [ ( ∂ u ∂ x 1 ) 2 + ⋯ + ( ∂ u ∂ x n ) 2 ] p − 2 2 {\displaystyle \quad |\nabla u|^{p-2}=\left[\textstyle \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 {\displaystyle 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 ( Ω ) {\displaystyle W^{1,p}(\Omega )} is a weak solution of
Δ p u = 0 in Ω {\displaystyle \Delta _{p}u=0{\mbox{ in }}\Omega }
if for every test function φ ∈ C 0 ∞ ( Ω ) {\displaystyle \varphi \in C_{0}^{\infty }(\Omega )} we have
∫ Ω | ∇ u | p − 2 ∇ u ⋅ ∇ φ d x = 0 {\displaystyle \int _{\Omega }|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \,dx=0}
where ⋅ {\displaystyle \cdot } denotes the standard scalar product.
Energy formulation The weak solution of the p-Laplace equation with Dirichlet boundary conditions
{ − Δ p u = f in Ω u = g on ∂ Ω {\displaystyle {\begin{cases}-\Delta _{p}u=f&{\mbox{ in }}\Omega \\u=g&{\mbox{ on }}\partial \Omega \end{cases}}}
in an open bounded set Ω ⊆ R N {\displaystyle \Omega \subseteq \mathbb {R} ^{N}} is the minimizer of the energy functional
J ( u ) = 1 p ∫ Ω | ∇ u | p d x − ∫ Ω f u d x {\displaystyle J(u)={\frac {1}{p}}\,\int _{\Omega }|\nabla u|^{p}\,dx-\int _{\Omega }f\,u\,dx}
among all functions in the Sobolev space W 1 , p ( Ω ) {\displaystyle W^{1,p}(\Omega )} satisfying the boundary conditions in the sense that u − g ∈ W 0 1 , p ( Ω ) {\displaystyle u-g\in W_{0}^{1,p}(\Omega )} (when Ω {\displaystyle \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 {\displaystyle f=1,g=0} and Ω {\displaystyle \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 | p p − 1 ) {\displaystyle u(x)=C\,\left(1-|x|^{\frac {p}{p-1}}\right)}
where C {\displaystyle C} is a suitable constant depending on the dimension N {\displaystyle N} and on p {\displaystyle p} only. Observe that for p > 2 {\displaystyle p>2} the solution is not twice differentiable in classical sense.
See also Infinity Laplacian
Notes
Sources Evans, Lawrence C. (1982). "A New Proof of Local C 1 , α {\displaystyle C^{1,\alpha }} 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 (3): 201–224. Bibcode:1977ArRMA..66..201L. doi:10.1007/bf00250671. MR 0477094. S2CID 120469946.
Further reading Ladyženskaja, O. A.; Solonnikov, V. A.; Ural'ceva, N. N. (1968), Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, Providence, RI: American Mathematical Society, pp. XI+648, ISBN 9780821886533, MR 0241821, Zbl 0174.15403. Uhlenbeck, K. (1977). "Regularity for a class of non-linear elliptic systems". Acta Mathematica. 138: 219–240. doi:10.1007/bf02392316. MR 0474389. Notes on the p-Laplace equation by Peter Lindqvist Juan Manfredi, Strong comparison Principle for p-harmonic functions