We obtain a solution to the [Formula: see text]-equation with exact support in a domain Ω with C1-smooth boundary satisfying property B in a complex manifold. This is done for complex-valued forms of type (r,s), s ≥ 1 and for forms of type (r,s), q ≤ s ≤ n - q, with values in a holomorphic vector bundle when the domain Ω is strongly q-convex.

Let Ω be a weakly q-convex domain in C n . We establish the L 2 existence theorem for the ∂-Neumann operator N when the boundary of Ω is C 1 . Using this result, we study the ∂ problem with exact support on such domains. Furthermore, there exists a number 0 > 0 such that the operators N , ∂ N and the Bergman projection are regular in the Sobolev space W (Ω) for < 0 when the boundary of Ω is C ∞ .

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This paper presents a systematic study of a mathematical model of glucose and insulin interaction with two time delays, with a focus on analytical studies, bifurcation analysis, and very well numerical simulations. This model based on the Intra-Venous Glucose Tolerance Test (IVGTT) and is presented with two time delays. One delay is the insulin response time to an increase in glucose concentration, and the hepatic glucose production time delay is the other. Then, we establish results on positivity, boundedness, and persistence. We also provide sufficient stability analysis conditions for both local and global asymptotic stability of the proposed models. For the latter, two different strategies are used: stability bifurcation analysis and Lyapunov-Krasovskii functionals. We investigate different regions of parameter space using two approaches, that yield different sets of sufficient conditions for global stability. The bifurcation graphs generated from our extensive and carefully designed simulations complement and confirm these analytical results. The insulin concentration level peaks after the glucose concentration level, according to the numerical simulations.