We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain D ⊂ ℝ^d and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by \({\cal P}\). We associate to Problem \({\cal P}\) an optimal control problem, denoted by \({\cal Q}\). Then, using appropriate Tykhonov triples, governed by a nonlinear operator G and a convex \(\tilde K\), we provide results concerning the well-posedness of problems \({\cal P}\) and \({\cal Q}\). 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 \({\cal P}\), constructed with particular choices of G and \(\tilde K\). 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.

我们考虑具有单边约束和亚微分边界条件的椭圆边界值问题。问题描述域中的传热d ⊂ℝ ^d和其弱制剂为半变分不等式的温度场，记的形式\（{\ CAL P} \） 。我们将问题\（{\ cal P} \）与最优控制问题相关联，以\（{\ cal Q} \）表示。然后，使用由非线性算子G和凸\（\ tilde K \）控制的适当的Tykhonov三元组，我们提供有关问题\（{\ cal P} \）和\（{\ cal Q } \）。我们的主要结果是定理4.2和5.2及其推论。他们的证明是基于紧密性，较低的半连续性和伪单调性的论据。此外，我们考虑了传热边值问题的三个相关扰动，这些扰动导致问题\（{\ cal P} \）的惩罚版本，该问题版本由G和\（\ tilde K \）的特定选择构成。我们证明定理4.2和5.2及其推论可以用于研究这些问题，以获得各种收敛结果。