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Many PDEs arise as the Euler-Lagrange equations of variational problems: they characterize critical points of energy functionals. The Dirichlet problem -Δu = f is equivalent to minimizing J(v) = ½∫|∇v|²dx - ∫fv dx over H¹₀(Ω). The direct method of the calculus of variations proves existence by showing that minimizing sequences converge (using weak compactness in Sobolev spaces and lower semicontinuity of the functional). Beyond simple minimization, variational methods include mountain pass theorems and minimax principles for finding saddle-point solutions of nonlinear PDEs.
Variational methods exploit the deep connection between PDEs and optimization. Many fundamental PDEs—including the equations of elasticity, electrostatics, minimal surfaces, and general relativity—are Euler-Lagrange equations of action or energy functionals. Solving the PDE is equivalent to finding critical points of the functional, and the rich machinery of optimization theory (convexity, compactness, topological methods) becomes available.
The direct method, developed by Tonelli and refined by subsequent generations, is the primary tool for proving existence of minimizers. The recipe is: (1) show the functional J is bounded below on the admissible set, (2) take a minimizing sequence {u_n} with J(u_n) → inf J, (3) extract a weakly convergent subsequence u_{n_k} ⇀ u (using the reflexivity of H¹ and the coercivity-induced boundedness), (4) show J(u) ≤ lim inf J(u_{n_k}) by weak lower semicontinuity. The minimizer u then satisfies the Euler-Lagrange equation, which is the PDE of interest.
For convex functionals, the direct method gives a unique global minimizer. The Dirichlet energy J(v) = ½∫|∇v|²dx - ∫fv dx is strictly convex, so the minimizer (= weak solution of -Δu = f) is unique. For nonlinear problems, the functional may be non-convex, possessing multiple critical points. The mountain pass theorem (Ambrosetti-Rabinowitz, 1973) handles a common non-convex geometry: if J has a local minimum at 0, and J(e) < J(0) for some distant point e, but J is large on a "mountain range" separating 0 from e, then there exists a critical point at the "mountain pass" level c = inf_{γ} max_{t∈[0,1]} J(γ(t)), where γ ranges over paths from 0 to e.
Beyond the mountain pass theorem, the landscape of variational methods includes the Ljusternik-Schnirelman theory (finding multiple critical points using topological index), the linking theorem, saddle point theorems, and concentration-compactness (handling loss of compactness for problems on unbounded domains). These methods have been spectacularly successful for nonlinear Schrodinger equations, semilinear elliptic problems, and geometric PDEs. The variational perspective also connects PDE theory to differential geometry (harmonic maps, minimal surfaces, Yamabe problem) and to physics (least action principle, Ginzburg-Landau theory), making it one of the most productive viewpoints in modern mathematics.