Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation. == Formal statement == Dirichlet's principle states that, if the function u ( x ) {\displaystyle u(x)} is the solution to Poisson's equation Δ u + f = 0 {\displaystyle \Delta u+f=0} on a domain Ω {\displaystyle \Omega } of R n {\displaystyle \mathbb {R} ^{n}} with boundary condition u = g {\displaystyle u=g} on the boundary ∂ Ω {\displaystyle \partial \Omega } , then u can be obtained as the minimizer of the Dirichlet energy E [ v ( x ) ] = ∫ Ω ( 1 2 | ∇ v | 2 − v f ) d x {\displaystyle E[v(x)]=\int _{\Omega }\left({\frac {1}{2}}|\nabla v|^{2}-vf\right)\,\mathrm {d} x} amongst all twice differentiable functions v {\displaystyle v} such that v = g {\displaystyle v=g} on ∂ Ω {\displaystyle \partial \Omega } (provided that there exists at least one function making the Dirichlet's integral finite).