Inverse problem for the Grad-Shafranov equation with affine right-hand side |
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Authors: | A S Demidov |
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Institution: | 1.Moscow State University and Moscow Institute of Physiscs and Technology,Moscow,Russia |
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Abstract: | For a given domain ω ⋐ ℝ2 with boundary γ = ∂ω, we study the cardinality of the set $
\mathfrak{A}_\eta \left( \Phi \right)
$
\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)
$
\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)
$
\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. |
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Keywords: | |
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