AThe boundary of the contact set where the solution touches the obstacle
BThe boundary of the domain Ω
CAny level set of the solution
DThe set where the solution is maximum
The obstacle problem seeks u ≥ ψ (above the obstacle ψ) that minimizes the Dirichlet energy. The solution coincides with ψ on the contact set and is harmonic (Δu = 0) elsewhere. The free boundary ∂{u = ψ} separates the contact region from the region where u lifts off the obstacle.
Question 2 True / False
The Stefan problem models the melting of ice, where the free boundary represents the ice-water interface.
TTrue
FFalse
Answer: True
In the Stefan problem, the temperature satisfies the heat equation in each phase (ice and water), and the interface moves according to the Stefan condition: the velocity of the interface equals the jump in heat flux across it. This conservation law ensures that the latent heat of fusion is accounted for.
Question 3 Short Answer
What regularity does the free boundary in the obstacle problem typically have?
Think about your answer, then reveal below.
Model answer: C^{1,α} (continuously differentiable with Holder continuous first derivatives) at regular points, with possible singular points
Caffarelli's fundamental work (1977-1998) showed that the free boundary in the obstacle problem is C^{1,α} near 'regular' points (where the solution grows quadratically away from the obstacle) and has a stratified structure at singular points. The singular set has Hausdorff dimension at most n-1.
Question 4 True / False
Free boundary problems can be reformulated as variational inequalities.
TTrue
FFalse
Answer: True
The obstacle problem is equivalent to finding u ∈ K = {v ∈ H¹₀ : v ≥ ψ} such that ∫∇u·∇(v-u)dx ≥ ∫f(v-u)dx for all v ∈ K. This variational inequality formulation avoids explicitly tracking the free boundary and is the basis for both theoretical analysis and numerical computation.