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Article

Sofonea, Mircea ; Tarzia, Domingo A.
Tykhonov well-posedness of a heat transfer problem with unilateral constraints. (English). Applications of Mathematics, vol. 67 (2022), issue 2, pp. 167-197
MSC: 35A16, 35M86, 49J20, 49J40, 49J45, 49J52, 80A19 | MR 4396683 | Zbl 07511500 | DOI: 10.21136/AM.2021.0172-20
Keywords:
heat transfer problem; unilateral constraint; subdifferential boundary condition; hemivariational inequality; optimal control; Tykhonov well-posedness; approximating sequence; convergence results
Summary:
We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D\subset \mathbb R^d$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $\mathcal P$. We associate to Problem $\mathcal P$ an optimal control problem, denoted by $\mathcal Q$. Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\widetilde {K}$, we provide results concerning the well-posedness of problems $\mathcal P$ and $\mathcal Q$. Our main results are Theorems 4.2 and 5.2, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem $\mathcal P$, constructed with particular choices of $G$ and $\widetilde {K}$. We prove that Theorems 4.2 and 5.2 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results.
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