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On optimal and quasi-optimal controls in coeﬃcients for multi-dimensional thermistor problem with mixed Dirichlet-Neumann boundary conditions

Kogut Peter I.

Control and Cybernetics

2019

Vol. 48, No. 1

31--68

In this paper we deal with an optimal control problem in coeﬃcients for the system of two coupled elliptic equations, also known as the thermistor problem, which provides a simultaneous description of the electric ﬁeld u = u(x) and temperature θ(x). The coeﬃcients of the operator div (B(x)∇θ(x)) are used as the controls in L∞(Ω). The optimal control problem is to minimize the discrepancy between a given distribution θd ∈ Lr(Ω) and the temperature of thermistor θ ∈ W1,γ 0 (Ω) by choosing an appropriate anisotropic heat conductivity matrix B. Basing on the perturbation theory of extremal problems and the concept of ﬁctitious controls, we propose an “approximation approach” and discuss the existence of the so-called quasi-optimal and optimal solutions to the given problem.

Bounded solutions for nonlinear elliptic equations in unbounded domains

Stańczy R.

Journal of Applied Analysis

2000

Vol. 6, nr 1

129-138

We prove here the existence of a bounded, radial solution in unbounded domain of the nonlinear elliptic problem ?u = f(||x||,u) for ||x||> 1, x ∈ Rn u(x)=0 for ||x||=1 under some asymptotic and sign condition on f. Under stronger assumptions it is proved that this solution must be of constant sign. The existence of radial solutions, vanishing at ? , of some semilinear equation is also established here.