The direct method of the calculus of variations proves existence by:
AShowing a minimizing sequence converges weakly to a minimizer
BConstructing an explicit solution formula
CApplying the maximum principle
DUsing the method of characteristics
The direct method takes a minimizing sequence {u_n} with J(u_n) → inf J, extracts a weakly convergent subsequence (using reflexivity and boundedness), and shows the limit is a minimizer (using weak lower semicontinuity of J). No explicit formula is needed.
Question 2 True / False
Weak lower semicontinuity of the functional J(u) = ∫|∇u|²dx means that if u_n ⇀ u weakly in H¹, then J(u) ≤ lim inf J(u_n).
TTrue
FFalse
Answer: True
This is the key property that ensures the limit of a minimizing sequence is actually a minimizer. The norm in a Hilbert space is weakly lower semicontinuous, and J(u) = ||∇u||² inherits this property. Without weak lower semicontinuity, the infimum might not be attained.
Question 3 Short Answer
What is the Euler-Lagrange equation associated with minimizing J(v) = ½∫|∇v|²dx - ∫fv dx?
Think about your answer, then reveal below.
Model answer: -Δu = f (Poisson's equation)
Setting the first variation δJ = 0 gives ∫∇u·∇φ dx = ∫fφ dx for all test functions φ, which is the weak form of -Δu = f. This shows that the Poisson equation is the gradient-flow equation for the Dirichlet energy.
Question 4 Multiple Choice
The mountain pass theorem is used to find:
ASaddle point (minimax) critical points of nonlinear functionals
BGlobal minimizers of convex functionals
CMaximum principles for elliptic equations
DEigenvalues of self-adjoint operators
When a functional has a local minimum and the energy is unbounded below along some direction, the mountain pass theorem guarantees a saddle point at the 'mountain pass' level. This is crucial for nonlinear elliptic equations where the functional is not convex and simple minimization is insufficient.