On the inverse problem for the Grad-Shafranov equation with affine right-hand side

For a given domain ω ⋐ ℝ2 with boundary γ = ∂ω, we study the cardinality of the set \( \mathfrak{A}_\eta \left( \Phi \right) \) of pairs of numbers (a, b) for which there is a function u = u (a,b): ω → ℝ such that ∇2 u(x) = au(x) + b ⩾ 0 for x ∈ ω, u| γ = 0, and ||∇u(s)| − Φ(s) ⩽ η for s ∈ γ. Here η ⩾ 0 stands for a very small number, Φ(s) = |∇(s)| / ∫ γ |∇v| d γ, and v is the solution of the problem ∇2 v = a 0 v + 1 ⩾ 0 on ω with v| γ = 0, where a 0 is a given number. The fundamental difference between the case η = 0 and the physically meaningful case η > 0 is proved. Namely, for η = 0, the set \( \mathfrak{A}_\eta \left( \Phi \right) \) contains only one element (a, b) for a broad class of domains ω, and a = a 0. On the contrary, for an arbitrarily small η > 0, there is a sequence of pairs (a j , b j ) ∈ \( \mathfrak{A}_\eta \left( \Phi \right) \) and the corresponding functions u j such that ‖f u j+1 ‖ − ‖f u j ‖ > 1, where ‖f u j = max x∈ω |f u j (x)| and f u j (x) = a j u j (x) + b j . Here the mappings f u j : ω → ℝ necessarily tend as j → ∞ to the δ-function concentrated on γ.