then u can be obtained as the minimizer of the Dirichlet's energy
amongst all twice differentiable functions such that on (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet.
Since the Dirichlet's integral is bounded from below, the existence of an infimum is guaranteed. That this infimum is attained was taken for granted by Riemann (who coined the term Dirichlet's principle) and others until Weierstrass gave an example of a functional that does not attain its minimum. Hilbert later justified Riemann's use of Dirichlet's principle.