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Definition

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The statement of the theorem

A partial differential equation is an equation that involves an unknown function of n≥2n\geq 2 variables and (some of) its partial derivatives. That is, for the unknown function u:U→R,u:U\rightarrow \mathbb {R} , of variables x=(x1,…,xn)x=(x_{1},\dots ,x_{n}) belonging to the open subset UU of Rn\mathbb {R} ^{n} , the kthk^{th} -order partial differential equation is defined as F[Dku,Dk−1u,…,Du,u,x]=0,F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0, where F:Rnk×Rnk−1⋯×Rn×R×U→R,F:\mathbb {R} ^{n^{k}}\times \mathbb {R} ^{n^{k-1}}\dots \times \mathbb {R} ^{n}\times \mathbb {R} \times U\rightarrow \mathbb {R} , and DD is the partial derivative operator. When writing PDEs, it is common to denote partial derivatives using subscripts. For example: ux=∂u∂x,uxx=∂2u∂x2,uxy=∂2u∂y ∂x=∂∂y(∂u∂x).u_{x}={\frac {\partial u}{\partial x}},\quad u_{xx}={\frac {\partial ^{2}u}{\partial x^{2}}},\quad u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right). In the general situation that u is a function of n variables, then u_{i} denotes the first partial derivative relative to the i-th input, u_{ij} denotes the second partial derivative relative to the i-th and j-th inputs, and so on. The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then Δu=u11+u22+⋯+unn.\Delta u=u_{11}+u_{22}+\cdots +u_{nn}. In the physics literature, the Laplace operator is often denoted by ∇^{2}; in the mathematics literature, ∇^{2}u may also denote the Hessian matrix of u. - ^Evans 1998, pp. 1–2.