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

We consider an elliptic boundary value problem with unilateral constraints and subdifferential conditions. The describes the heat transfer in a domain $D\subset\R^d$ its weak formulation is form of hemivariational inequality for temperature field, denoted by $\cP$. associate to Problem $\cP$ optimal control problem, $\cQ$. Then, using appropriate Tykhonov triples, governed nonlinear operator $G$ convex $\wK$, we provide results concerning well-posedness problems Our main are Theorems 14 18, together their corollaries. Their proofs based on arguments compactness, lower semicontinuity pseudomonotonicity. Moreover, three relevant perturbations valued which lead penalty versions $\cP$, constructed particular choices $\wK$. prove that 18 as well corollaries can be applied study these problems, order obtain various convergence results.