Variational Methods — Mathematics Knowledge System

Overview

Variational methods reformulate boundary value problems as minimisation (or saddle-point) problems for energy functionals on Sobolev spaces. The key idea is that solutions to a PDE like -Delta u = f can be characterised as the minimiser of the energy functional E(u) = (1/2)integral|grad u|^2 - f u over an appropriate Sobolev space. The Lax–Milgram theorem and coercivity provide the functional-analytic tools guaranteeing existence and uniqueness of the weak solution.

Intuition

Physical equilibria are often energy minimisers: a stretched elastic membrane finds its shape by minimising elastic energy, a conducting solid reaches thermal equilibrium by minimising heat energy. Variational methods make this rigorous in infinite dimensions: replace the PDE by 'find u minimising E(u)', use Sobolev spaces as the domain, and apply existence theory from functional analysis. The weak formulation (find u with a(u,v) = L(v) for all test functions v) is the Euler–Lagrange equation of the minimisation problem.

Formal Definition

Definition

Let V be a Hilbert space (typically a Sobolev space H^1_0(Omega)). A bilinear form a: V x V -> R is coercive if a(u,u) >= alpha ||u||^2 for some alpha > 0, and continuous (bounded) if |a(u,v)| <= M ||u|| ||v||. The Lax–Milgram theorem says: if a is continuous and coercive and L: V -> R is a continuous linear functional, then there exists a unique u in V with a(u,v) = L(v) for all v in V. This u is the unique minimiser of E(u) = (1/2)a(u,u) - L(u).

E(u) = \frac{1}{2}a(u,u) - L(u),\quad u \in V

Energy functional

a(u,v) = L(v) \quad \forall v \in V

Weak formulation (Euler–Lagrange)

a(u,u) \ge \alpha\|u\|^2 \quad (\text{coercivity}),\quad |a(u,v)| \le M\|u\|\|v\| \quad (\text{continuity})

Lax–Milgram hypotheses

Notation

Notation	Meaning
$H^1_0(\Omega)$	Sobolev space with zero boundary trace
$a(u,v)$	Bilinear form associated to the PDE
$L(v)$	Linear functional (loading)
$\alpha$	Coercivity constant

Theorems

Theorem 11.1: Theorem 11.1 (Lax–Milgram)

Let V be a Hilbert space, a: V x V -> R continuous and coercive, L: V -> R continuous linear. Then there exists a unique u in V with a(u,v) = L(v) for all v in V, and the map L |-> u is continuous: ||u|| <= ||L||/alpha.

Theorem 11.2: Theorem 11.2 (Dirichlet problem)

Let Omega be a bounded open set in R^n and f in L^2(Omega). The bilinear form a(u,v) = integral_Omega grad u . grad v is coercive on H^1_0(Omega) by the Poincaré inequality. Hence there exists a unique u in H^1_0(Omega) with integral grad u . grad v = integral fv for all v in H^1_0(Omega), i.e., a unique weak solution to -Delta u = f with zero Dirichlet boundary conditions.

Theorem 11.3: Theorem 11.3 (Mountain Pass)

For C^1 functionals E: V -> R that are not bounded below but satisfy the Palais–Smale condition, and have a local minimum and another point where E is lower, the mountain pass theorem guarantees a critical point (saddle point) of E.

Worked Examples

1
Set V = H^1(R^n) with norm ||u||^2 = integral(|grad u|^2 + u^2). Define a(u,v) = integral(grad u . grad v + uv).
2
Continuity: |a(u,v)| <= ||u||_{H^1} ||v||_{H^1} by Cauchy–Schwarz.
3
Coercivity: a(u,u) = ||u||_{H^1}^2 >= 1 * ||u||^2_{H^1}. So alpha = 1.
4
L(v) = integral fv is continuous on H^1 by Cauchy–Schwarz: |L(v)| <= ||f||_{L^2} ||v||_{L^2} <= ||f||_{L^2} ||v||_{H^1}.
5
Lax–Milgram gives unique u in H^1 with a(u,v) = L(v) for all v, i.e., integral(grad u . grad v + uv) = integral fv, the weak form of -Delta u + u = f.

✓ Answer

By Lax–Milgram with coercivity constant alpha = 1, the equation -Delta u + u = f has a unique weak solution u in H^1(R^n) for every f in L^2(R^n).

Practice Problems

Mediumproof writing

Show that the minimiser of E(u) = (1/2)a(u,u) - L(u) over a Hilbert space V satisfies the Euler–Lagrange equation a(u,v) = L(v) for all v in V.

Hardfree response

Explain the role of coercivity in guaranteeing that minimising sequences for E(u) = (1/2)a(u,u) - L(u) are bounded.

Quiz

Summary

Variational methods reformulate PDE boundary value problems as energy minimisation over Sobolev spaces.
Lax–Milgram theorem: continuous coercive a with continuous L gives unique u with a(u,v) = L(v) for all v.
The minimiser of E(u) = (1/2)a(u,u) - L(u) satisfies the Euler–Lagrange equation.
Poincaré inequality for bounded domains gives coercivity of the Dirichlet form.
Non-linear problems require more advanced tools: mountain pass theorem, minimax theory.

References

BookBrezis, H. — Functional Analysis, Sobolev Spaces and PDEs, Chapter 9
BookEvans, L. — Partial Differential Equations (2nd ed.), Chapter 6
Websiteen.wikipedia.org