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

Mircea Sofonea; Domingo A. Tarzia

Applications of Mathematics (2022)

  • Volume: 67, Issue: 2, page 167-197
  • ISSN: 0862-7940

Abstract

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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 𝒫 . We associate to Problem 𝒫 an optimal control problem, denoted by 𝒬 . Then, using appropriate Tykhonov triples, governed by a nonlinear operator G and a convex K ˜ , we provide results concerning the well-posedness of problems 𝒫 and 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 𝒫 , constructed with particular choices of G and 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.

How to cite

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Sofonea, Mircea, and Tarzia, Domingo A.. "Tykhonov well-posedness of a heat transfer problem with unilateral constraints." Applications of Mathematics 67.2 (2022): 167-197. <http://eudml.org/doc/297926>.

@article{Sofonea2022,
abstract = {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.},
author = {Sofonea, Mircea, Tarzia, Domingo A.},
journal = {Applications of Mathematics},
keywords = {heat transfer problem; unilateral constraint; subdifferential boundary condition; hemivariational inequality; optimal control; Tykhonov well-posedness; approximating sequence; convergence results},
language = {eng},
number = {2},
pages = {167-197},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Tykhonov well-posedness of a heat transfer problem with unilateral constraints},
url = {http://eudml.org/doc/297926},
volume = {67},
year = {2022},
}

TY - JOUR
AU - Sofonea, Mircea
AU - Tarzia, Domingo A.
TI - Tykhonov well-posedness of a heat transfer problem with unilateral constraints
JO - Applications of Mathematics
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 167
EP - 197
AB - 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.
LA - eng
KW - heat transfer problem; unilateral constraint; subdifferential boundary condition; hemivariational inequality; optimal control; Tykhonov well-posedness; approximating sequence; convergence results
UR - http://eudml.org/doc/297926
ER -

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