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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 Where the | ∇ u | p − 2 {\displaystyle |\nabla 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 {\displaystyle W^{1,p}} is a weak solution of if for every test function φ ∈ C 0 ∞ {\displaystyle \varphi \in C_{0}^{\infty }} we have