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 {displaystyle p} is allowed to range over 1 < p < ∞ {displaystyle 1<p<infty } . It is written as 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 Where the | ∇ u | p − 2 {displaystyle | abla u|^{p-2}} is defined as 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 if for every test function φ ∈ C 0 ∞ ( Ω ) {displaystyle varphi in C_{0}^{infty }(Omega )} we have where ⋅ {displaystyle cdot } denotes the standard scalar product. The weak solution of the p-Laplace equation with Dirichlet boundary conditions in a domain Ω ⊂ R N {displaystyle Omega subset mathbb {R} ^{N}} is the minimizer of the energy functional among all functions in the Sobolev space W 1 , p ( Ω ) {displaystyle W^{1,p}(Omega )} satisfying the boundary conditions in the 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 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.