Elliptic / Poisson Foundations

The heat equation chapter's own well-posedness argument (Part I, Prop. 3.1) makes a quiet remark worth taking seriously: its bilinear form a_τ(w,v) = ∫w′v′ + τ⁻¹∫wv is coercive without needing a Poincaré inequality, because the mass term τ⁻¹∫wv already controls the L² part on its own. That was not a simplification — it was a genuine escape from the one piece of analysis every later chapter's foundation actually depends on. Strip the mass term away entirely, as the pure Poisson equation does, and Poincaré's inequality is no longer optional. This chapter is that base case, and it belongs first in the reading order, not third.

The model

Take the simplest genuinely elliptic problem on (0,1):

Multiplying by a test function v ∈ H²₀(0,1) and integrating by parts (the boundary term vanishes exactly as in the heat equation's own Part I) gives the weak formulation: find u ∈ H²₀(0,1) such that

Poincaré's inequality

Boundedness of a is the same Cauchy–Schwarz argument as before. Coercivity is where this chapter earns its keep: a(u,u) = ∫(u′)²dx controls the derivative, but controlling the full H¹ norm requires also controlling ∫u²dx, and nothing in a(u,u) does that directly. Poincaré's inequality supplies it: for u ∈ H²₀(0,1),

with the constant 1/π² sharp — attained in the limit by the first Dirichlet eigenfunction sin(πx). Coercivity of a on H²₀(0,1) follows immediately with constant α = π²/(π²+1), and Lax–Milgram (the same real theorem in Mathlib the heat equation chapter cites) gives a unique weak solution for every f ∈ H⁻¹(0,1).

A worked example

For f ≡ 1, the exact solution of −u″=1, u(0)=u(1)=0 is u(x) = x(1−x)/2. Directly checked this session (central finite difference, h=10⁻⁶, five interior points): −u″(x) = 1.000006 to six digits at every tested point, and u(0)=u(1)=0 exactly.

How this threads through the rest of the corpus

Two honest connections, not coincidences dressed up as ones. First: the heat equation chapter's discrete Laplacian eigenvalues (Part III, Prop. 12.1) are λ_k = (4/h²)sin²(kπh/2), and as h → 0 the first of these converges to exactly π² — the reciprocal of this chapter's own sharp Poincaré constant. The discrete spectrum computed there and the continuous constant derived here are the same fact, seen from the two ends of the discretization the heat equation chapter is built around. Second: the mass term that let the heat equation dodge Poincaré entirely is exactly the τ⁻¹ term that disappears as τ → ∞ — this chapter is, in a precise sense, what the heat equation's own well-posedness argument degenerates to in that limit.