Weak solutions via two-field Lagrange multipliers for boundary value problems in mathematical physics
Abstract
A new variational approach for a boundary value problem in mathematical physics is proposed. By considering two-field Lagrange multipliers, we deliver a variational formulation consisting of a mixed variational problem which is equivalent with a saddle point problem. Thus, the unique solvability of the weak formulation we propose is governed by the saddle point theory. Alternative variational formulations and some of their connections are also discussed.
Keyword : partial differential equations, subdifferential inclusions, two-field Lagrange multipliers, weak solutions, saddle points
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
H. Brézis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2010. https://doi.org/10.1007/978-0-387-70914-7
F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. SpringerVerlag, New York, 1991. https://doi.org/10.1007/978-1-4612-3172-1
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. Saddle point formulations for a class of nonlinear boundary value problems. Bull. Math. Soc. Sci. Math. Roumanie, 64(112):355–368, 2021.
L. Debnath and P. Mikusiński. Introduction to Hilbert spaces with Applications3rd Edition. Elsevier Academic Press, 2005.
I. Ekeland and R. Témam. Convex Analysis and Variational Problems. Classics in Applied Mathematics, 28, SIAM, Philadelphia, PA, 1999. https://doi.org/10.1137/1.9781611971088
J. Haslinger, I. Hlaváček and J. Nečas. Numerical methods for unilateral problems in solid mechanics. In Finite Element Methods (Part 2), Numerical Methods for Solids (Part 2), volume 4 of Handbook of Numerical Analysis, pp. 313–485. Elsevier, 1996. https://doi.org/10.1016/S1570-8659(96)80005-6
S. Hüeber, A. Matei and B. Wohlmuth. Efficient algorithms for problems with friction. SIAM Journal on Scientific Computing, 29(1):70–92, 2007. https://doi.org/10.1137/050634141
A. Kovetz. Electromagnetic theory. Oxford University Press, 2000.
A. Kufner, O. John and S. Fucik. Function Spaces, in: Monographs and Textbooks on Mechanics of Solids and Fluids. Noordhoff International Publishing, Leyden, 1977.
A. Matei. On the relationship between alternative variational formulations of a frictional contact model. Journal of Mathematical Analysis and Applications, 480(1):123391, 2019. https://doi.org/10.1016/j.jmaa.2019.123391
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
P. Monk. Finite Element Methods for Maxwell’s Equations. Oxford University Press, 2003. https://doi.org/10.1093/acprof:oso/9780198508885.001.0001
J. Muscat. Hilbert Spaces, pp. 171–219. Springer International Publishing, Cham, 2014. ISBN 978-3-319-06728-5. https://doi.org/10.1007/978-3-319-06728-5_10
J. Neˇcas. Direct Methods in the Theory of Elliptic Equations. Springer, 2012. https://doi.org/10.1007/978-3-642-10455-8
M. Sofonea and A. Matei. Evolutionary Variational Inequalities, pp. 1– 34. Springer New York, New York, NY, 2009. ISBN 978-0-387-87460-9. https://doi.org/10.1007/978-0-387-87460-9_5
M. Sofonea and A. Matei. Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series. Cambridge University Press, 2012. https://doi.org/10.1017/CBO9781139104166