Thermistor Problem: Multi-Dimensional Modelling, Optimization And Approximation

Abstract

We consider a problem of an optimal control in coefficients for the system of two coupled elliptic equations also known as thermistor problem which provides a simultaneous description of the electric field u = u(x) and temperature θ(x). The coefficients of operator div (A(x) ∇ θ(x)) are used as the controls in L ∞ (Ω). The optimal control problem is to minimize the discrepancy between a given distribution θ d ∈ L r (Ω) and the temperature of thermistor θ ∈ W 1,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.

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing

“…Since the last inequality is valid for all ϕ ∈ C ∞ 0 (R N ) and C ∞ 0 (R N ) is a dense subset of L p (•) ( ), it follows that this relation also holds true for ϕ ∈ L p (•) ( ). So, taking ϕ = | f (x)| p(x)−2 f (x), we arrive at the announced inequality (21). As a consequence of (21) and estimate (A4), we get:…”

“…So, taking ϕ = | f (x)| p(x)−2 f (x), we arrive at the announced inequality (21). As a consequence of (21) and estimate (A4), we get:…”

“…it can be shown that the lower semicontinuity property ( 21) can be generalized as follows: (26) Following in some technical aspects the recent studies [21,22,30], we give the following existence result.…”

“…Thus, in spite of the fact that in the literature there are many approaches to the study of variational problems in abstract functional spaces, the above-mentioned circumstances make the problem (10) challenging (see [9,18,19,21,22,30] for the recent studies in this field).…”