On the inverse doping profile problem

Here V is electrostatic potential, u, v are Slotboom concentrations of electrons and holes, λ = 10−3 − 10−6, δ = ni (intrinsic density of electrons), C ∈ L∞(Ω) is doping profile, Q is recombination-regeneration rate, q is elementary charge, μn, μp are mobilities of electrons and holes, and Vbi, U are equilibrium and applied potentials. We consider Ω = (0, 1) × (−h, 0) in R (h is some positive number), Γ1 = (0, l) × {0}, 0 < l < 1, Γ0 = (0, 1) × {−h}, ΓN = ∂Ω \ (Γ0 ∪ Γ1). ν is (outer) unit normal. We assume Q ∈ C, 0 ≤ Q and ‖Q(, V, u, v)‖∞(Ω)+‖∇V,u,vQ(, V, u, v)‖∞(Ω) ≤ C(M) when |V |+|u|+|v| ≤M , μp, μn ∈ C, 0 ≤ μp, 0 ≤ μn, and U, Vbi ∈ H2(Ω)∩L∞(Ω). By using Fixed Point Theorems one can show existence of a weak (H1(Ω)∩L∞(Ω)) solution to (1.1), (1.2) and its uniqueness for small U [3], [4], [12].