Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Sofonea, Mircea ; Tarzia, Domingo A.

Applications of Mathematics, Tome 67 (2022) no. 2, pp. 167-197

Résumé

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.

Détail

DOI : 10.21136/AM.2021.0172-20

Classification : 35A16, 35M86, 49J20, 49J40, 49J45, 49J52, 80A19
Keywords: heat transfer problem; unilateral constraint; subdifferential boundary condition; hemivariational inequality; optimal control; Tykhonov well-posedness; approximating sequence; convergence results