Variational methods solve differential equations by identifying solutions as minimizers or critical points of an energy. Compactness supplies a limit, lower semicontinuity preserves the energy inequality, and differentiation of the functional recovers the equation.
The direct method
Theorem (direct method). Let \(X\) be a reflexive Banach space and \(J:X\to(-\infty,\infty]\) be proper, coercive, and sequentially weakly lower semicontinuous. Then \(J\) attains its minimum.
Proof. Choose a minimizing sequence \((u_k)\). Coercivity makes it bounded. Reflexivity gives a weakly convergent subsequence \(u_{k_j}\rightharpoonup u\). Weak lower semicontinuity yields \[ J(u)\le\liminf_{j\to\infty}J(u_{k_j})=\inf_XJ, \] so \(u\) is a minimizer.
Example (Poisson equation). On \(H_0^1(\Omega)\), define \[ J(u)=\frac12\int_\Omega|\nabla u|^2\,\mathrm{d}x -\int_\Omega fu\,\mathrm{d}x. \] Poincar?'s inequality makes \(J\) coercive, convexity makes it weakly lower semicontinuous, and the direct method gives a minimizer. For every \(v\in H_0^1(\Omega)\), \[ 0=\left.\frac{\mathrm{d}}{\mathrm{d}t}J(u+tv)\right|_{t=0} =\int_\Omega\nabla u\cdot\nabla v\,\mathrm{d}x-\int_\Omega fv\,\mathrm{d}x, \] which is precisely the weak form of \(-\Delta u=f\).
Euler-Lagrange equations
Proposition. If \[ J(u)=\int_\Omega L(x,u,\nabla u)\,\mathrm{d}x \] and \(u\) is a smooth local minimizer with fixed boundary values, then \[ L_z(x,u,\nabla u) -\operatorname{div}L_p(x,u,\nabla u)=0. \]
Proof. For \(\varphi\in C_c^\infty(\Omega)\), differentiate \(J(u+t\varphi)\) at \(t=0\): \[ 0=\int_\Omega \bigl(L_z\varphi+L_p\cdot\nabla\varphi\bigr)\,\mathrm{d}x =\int_\Omega \bigl(L_z-\operatorname{div}L_p\bigr)\varphi\,\mathrm{d}x. \] The fundamental lemma of the calculus of variations gives the equation.
Strict convexity of \(J\) implies uniqueness of the minimizer. Convexity also turns the first-order condition into a global one: \[ 0\in\partial J(u) \quad\Longleftrightarrow\quad J(v)\ge J(u)\ \text{for all }v. \]
Eigenvalues through constrained minimization
Theorem (first Dirichlet eigenvalue). For a bounded domain, \[ \lambda_1=\min_{\substack{u\in H_0^1(\Omega)\\u\ne0}} \frac{\int_\Omega|\nabla u|^2\,\mathrm{d}x} {\int_\Omega|u|^2\,\mathrm{d}x} \] is attained, and a minimizer satisfies \[ -\Delta u=\lambda_1u \] weakly. It may be chosen positive and \(\lambda_1\) is simple on a connected domain.
Proof. Normalize a minimizing sequence by \(\|u_k\|_2=1\). It is bounded in \(H_0^1\), hence converges weakly there and, by Rellich compactness, strongly in \(L^2\). The limit retains unit norm and minimizes by weak lower semicontinuity. A Lagrange multiplier gives the eigenvalue equation. Positivity follows by replacing \(u\) with \(|u|\); the strong maximum principle and the standard quotient argument give strict positivity and simplicity.
Critical points beyond minima
Theorem (mountain-pass principle). Let \(J\in C^1(X,\mathbb R)\) satisfy the Palais-Smale compactness condition. Suppose \(J(0)=0\), \(J\ge\alpha>0\) on \(\|u\|=r\), and \(J(e)<0\) for some \(\|e\|>r\). Then \[ c=\inf_{\gamma(0)=0,\,\gamma(1)=e}\max_{t\in[0,1]}J(\gamma(t)) \] is a critical value of \(J\), with \(c\ge\alpha\).
The geometry forces every path from \(0\) to \(e\) over an energy barrier. The Palais-Smale condition prevents an almost-critical sequence at that height from escaping or oscillating without a convergent subsequence. Semilinear elliptic equations often possess both a minimizing solution and a second mountain-pass solution.
Gradient flows and compactness defects
The evolution \[ u_t\in-\partial J(u) \] is the gradient flow of \(J\). Formally, \[ \frac{\mathrm{d}}{\mathrm{d}t}J(u(t)) =-\|u_t(t)\|^2\le0. \] Heat flow is the \(L^2\)-gradient flow of Dirichlet energy; nonlinear diffusions arise from other convex energies and metrics.
On unbounded domains or at critical Sobolev exponents, bounded minimizing sequences need not be compact. They may translate, vanish, or split into separated packets. Concentration-compactness isolates these alternatives and replaces a false compactness assertion with a precise analysis of the lost mass.