On a class of saddle point problems and convergence results
Abstract
We consider an abstract mixed variational problem consisting of two inequalities. The first one is governed by a functional φ, possibly non-differentiable. The second inequality is governed by a nonlinear term depending on a non negative parameter ǫ. We study the existence and the uniqueness of the solution by means of the saddle point theory. In addition to existence and uniqueness results, we deliver convergence results for ǫ → 0. Finally, we illustrate the abstract results by means of two examples arising from contact mechanics.
Keyword : mixed variational problem, penalty term, saddle point, convergence result
How to Cite
Chivu Cojocaru, M., & Matei, A. (2020). On a class of saddle point problems and convergence results. Mathematical Modelling and Analysis, 25(4), 608-621. https://doi.org/10.3846/mma.2020.11140
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
R.A. Adams. Sobolev spaces. Academic Press, 1975.
Y. Bai, S. Migorski and S. Zeng. Well-posedness of a class of generalized mixed hemivariational-variational inequalities. Nonlinear Analysis: Real World Applications, 48:424–444, 2019. https://doi.org/10.1016/j.nonrwa.2019.02.001
D. Braess. Finite elements. Theory, solvers, and applications in solid mechanics. Cambridge University Press, 2007. https://doi.org/10.1017/CBO9780511618635
H. Brézis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2010. https://doi.org/10.1007/978-0-387-70914-7
R. Ciurcea and A. Matei. Solvability of a mixed variational problem. Ann. Univ. Craiova, 36(1):105–111, 2009.
D. Cohen. An Introduction to Hilbert Space and Quantum Logic. Springer-Verlag New York, 1989. https://doi.org/10.1007/978-1-4613-8841-8
M. Chivu Cojocaru and A. Matei. Well-posedness for a class of frictional contact models via mixed variational formulations. Nonlinear Analysis: Real World Applications, 47:127–141, 2019. https://doi.org/10.1016/j.nonrwa.2018.10.009
L. Debnath and P. Mikusiński. Introduction to Hilbert spaces with Applications-3rd Edition. Elsevier Academic Press, 2005.
I. Ekeland and R. Témam. Convex Analysis and Variational Problems. Classics in Applied Mathematics, SIAM, 28, 1999. https://doi.org/10.1137/1.9781611971088
J. Haslinger, I. Hlaváček and J. Nečas. Numerical methods for unilateral problems in solid mechanics, in: P.G. Ciarlet, J.-L. Lions (Eds.), Handbook of Numerical Analysis, Vol. IV, 313–485. North-Holland, Amsterdam, 1996. https://doi.org/10.1016/S1570-8659(96)80005-6
A. Matei. Weak solvability via Lagrange multipliers for contact problems involving multi-contact zones. Mathematics and Mechanics of Solids, 21(7):826–841, 2016. https://doi.org/10.1177/1081286514541577
A. Matei. A mixed hemivariational-variational problem and applications. Computers and Mathematics with Applications, 77(11):2989–3000, 2019. https://doi.org/10.1016/j.camwa.2018.08.068
A. Matei, S. Sitzmann, K. Willner and B. Wohlmuth. A mixed variational formulation for a class of contact problems in viscoelasticity. Applicable Analysis, 97(8):1340–1356, 2018. https://doi.org/10.1080/00036811.2017.1359569
A. Matei and M. Sofonea. A mixed variational formulation for a piezoelectric frictional contact problem. IMA Journal of Applied Mathematics, 82(2):334–354, 2017.
S. Migorski, Y. Bai and S. Zeng. A class of generalized mixed variational-hemivariational inequalities II: Applications. Nonlinear Analysis: Real World Applications, 50:633–650, 2019. https://doi.org/10.1016/j.nonrwa.2019.06.006
S. Migorski, A. Ochal and M. Sofonea. Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Springer, 2013. https://doi.org/10.1007/978-1-4614-4232-5
M. Sofonea and A. Matei. Mathematical Models in Contact Mechanics. Cambridge University Press, 2012.
M. Sofonea, A. Matei and Y. Xiao. Optimal control for a class of mixed variational problems. Zeitschrift für angewandte Mathematik und Physik, 70(4):127, 2019. https://doi.org/10.1007/s00033-019-1173-4
M. Sofonea, Y.B. Xiao and M. Couderc. Optimization problems for a viscoelastic frictional contact problem with unilateral constraints. Nonlinear Analysis: Real World Applications, 50:86–103, 2019. https://doi.org/10.1016/j.nonrwa.2019.04.005
Y.B. Xiao and M. Sofonea. Generalized penalty method for elliptic variational hemivariational inequalities. Appl Math Optim, 2019. https://doi.org/10.1007/s00245-019-09563-4
Y. Bai, S. Migorski and S. Zeng. Well-posedness of a class of generalized mixed hemivariational-variational inequalities. Nonlinear Analysis: Real World Applications, 48:424–444, 2019. https://doi.org/10.1016/j.nonrwa.2019.02.001
D. Braess. Finite elements. Theory, solvers, and applications in solid mechanics. Cambridge University Press, 2007. https://doi.org/10.1017/CBO9780511618635
H. Brézis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2010. https://doi.org/10.1007/978-0-387-70914-7
R. Ciurcea and A. Matei. Solvability of a mixed variational problem. Ann. Univ. Craiova, 36(1):105–111, 2009.
D. Cohen. An Introduction to Hilbert Space and Quantum Logic. Springer-Verlag New York, 1989. https://doi.org/10.1007/978-1-4613-8841-8
M. Chivu Cojocaru and A. Matei. Well-posedness for a class of frictional contact models via mixed variational formulations. Nonlinear Analysis: Real World Applications, 47:127–141, 2019. https://doi.org/10.1016/j.nonrwa.2018.10.009
L. Debnath and P. Mikusiński. Introduction to Hilbert spaces with Applications-3rd Edition. Elsevier Academic Press, 2005.
I. Ekeland and R. Témam. Convex Analysis and Variational Problems. Classics in Applied Mathematics, SIAM, 28, 1999. https://doi.org/10.1137/1.9781611971088
J. Haslinger, I. Hlaváček and J. Nečas. Numerical methods for unilateral problems in solid mechanics, in: P.G. Ciarlet, J.-L. Lions (Eds.), Handbook of Numerical Analysis, Vol. IV, 313–485. North-Holland, Amsterdam, 1996. https://doi.org/10.1016/S1570-8659(96)80005-6
A. Matei. Weak solvability via Lagrange multipliers for contact problems involving multi-contact zones. Mathematics and Mechanics of Solids, 21(7):826–841, 2016. https://doi.org/10.1177/1081286514541577
A. Matei. A mixed hemivariational-variational problem and applications. Computers and Mathematics with Applications, 77(11):2989–3000, 2019. https://doi.org/10.1016/j.camwa.2018.08.068
A. Matei, S. Sitzmann, K. Willner and B. Wohlmuth. A mixed variational formulation for a class of contact problems in viscoelasticity. Applicable Analysis, 97(8):1340–1356, 2018. https://doi.org/10.1080/00036811.2017.1359569
A. Matei and M. Sofonea. A mixed variational formulation for a piezoelectric frictional contact problem. IMA Journal of Applied Mathematics, 82(2):334–354, 2017.
S. Migorski, Y. Bai and S. Zeng. A class of generalized mixed variational-hemivariational inequalities II: Applications. Nonlinear Analysis: Real World Applications, 50:633–650, 2019. https://doi.org/10.1016/j.nonrwa.2019.06.006
S. Migorski, A. Ochal and M. Sofonea. Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Springer, 2013. https://doi.org/10.1007/978-1-4614-4232-5
M. Sofonea and A. Matei. Mathematical Models in Contact Mechanics. Cambridge University Press, 2012.
M. Sofonea, A. Matei and Y. Xiao. Optimal control for a class of mixed variational problems. Zeitschrift für angewandte Mathematik und Physik, 70(4):127, 2019. https://doi.org/10.1007/s00033-019-1173-4
M. Sofonea, Y.B. Xiao and M. Couderc. Optimization problems for a viscoelastic frictional contact problem with unilateral constraints. Nonlinear Analysis: Real World Applications, 50:86–103, 2019. https://doi.org/10.1016/j.nonrwa.2019.04.005
Y.B. Xiao and M. Sofonea. Generalized penalty method for elliptic variational hemivariational inequalities. Appl Math Optim, 2019. https://doi.org/10.1007/s00245-019-09563-4