The Hopf lemma states that if u is a supersolution with u(x₀) = max u at a boundary point x₀, then:
A∂u/∂ν(x₀) > 0, where ν is the outward normal
Bu is constant in a neighborhood of x₀
C∇u(x₀) = 0
Du has a saddle point at x₀
The Hopf lemma gives a quantitative version of the strong maximum principle at the boundary: if u achieves its maximum at a boundary point, it must be increasing in the inward normal direction there. The normal derivative is strictly positive (assuming an interior sphere condition at x₀).
Question 2 True / False
Harnack's inequality for positive harmonic functions states sup u ≤ C inf u on compact subsets.
TTrue
FFalse
Answer: True
For a positive harmonic function u in a domain Ω, on any compact subdomain K ⊂ Ω, sup_K u ≤ C inf_K u where C depends only on K and Ω. This is a remarkably strong estimate: the maximum and minimum of a positive solution are comparable.
Question 3 Short Answer
What is the ABP (Alexandrov-Bakelman-Pucci) maximum principle?
Think about your answer, then reveal below.
Model answer: A quantitative L^∞ bound: sup u ≤ sup_{∂Ω} u⁺ + C·diam(Ω)·||f||_{Lⁿ(Ω)} for Lu ≥ -f with f ≥ 0
The ABP estimate controls the maximum of a subsolution by its boundary values and the Lⁿ norm of the right-hand side (not L^∞, which is much stronger). The proof uses the contact set and the area formula. This estimate is the starting point for the Krylov-Safonov regularity theory.
Question 4 True / False
The strong maximum principle requires the operator to satisfy an interior sphere condition at boundary points.
TTrue
FFalse
Answer: False
The strong maximum principle is an interior result: if u achieves its maximum at an interior point, u is constant. The interior sphere condition is needed for the Hopf lemma (a boundary result). The strong maximum principle holds for general elliptic operators on any connected domain.