Dirichlet's Principle

Dirichlet's Principle states that if you want to find a function that minimizes a certain energy-like quantity (called an integral), subject to specific boundary conditions, such such a function exists and is unique. In simple terms, it assures that among all possible ways to smoothly fill in a shape's interior while matching given values on the boundary, there is a single "best" way that minimizes energy or distortion. This principle is fundamental in solving various physical and mathematical problems, like finding steady-state heat distributions or electrical potentials, by turning boundary conditions into an optimal solution inside.