Abstract

An obstacle problem with a memory term is studied in the framework of the variational inequality theory. Applying a fixed point argument and a convergence result for convex sets the existence and uniqueness of a solution are proved. Regularity results with respect to time and space are then deduced by using a penalization method. Furthermore, the time evolution of the solution is discussed. The particular evolutionary variational inequalities result from the application of a generalized Baiocchi-type transformation to degenerate free boundary problems with space- and, in particular, time-dependent coefficients. One of the applications is given by a quasi-stationary model for the electro-chemical machining problem.

Author information

Contact details are reproduced from the original publication and may be historical.

Jörg Steinbach

Chair of Applied Mathematics, University of Technology, Center for Mathematical Sciences, Arcisstr. 21, 80290 Munich, Germany

Keywords

  • Evolutionary variational inequality
  • existence
  • uniqueness
  • regularity and time evolution of the solution
  • convex sets
  • penalization method
  • electro-chemical machining process

References

  1. M. Chipot: Variational Inequalities and Flow in Porous Media, Springer-Verlag, Berlin et al., 1984.https://doi.org/10.1007/978-1-4612-1120-4Google Scholar
  2. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland Publishing Company, Amsterdam et al., 1976.Google Scholar
  3. C. M. Elliott: On a variational inequality formulation of an electrochemical machining moving boundary problem and its approximation by the finite element method, Journ. Inst. Math. Appl. (IMA) 25 (1980) 121-131.https://doi.org/10.1093/imamat/25.2.121Google Scholar
  4. H. Vogel: Gerthsen, Physik, Springer-Verlag, Berlin et al., 1995.https://doi.org/10.1007/978-3-662-07466-4Google Scholar
  5. R. Glowinski: Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, Berlin et al., 1984.https://doi.org/10.1007/978-3-662-12613-4Google Scholar
  6. P. Grisvard: Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Program, Boston et al., 1985.Google Scholar
  7. K.-H. Hoffmann, J. Sprekels (eds.): Free Boundary Problems: Theory and Applications, Pitman Research Notes in Mathematics Series 185, 186, 1990.https://doi.org/10.1007/978-3-0348-7301-7Google Scholar
  8. D. Kinderlehrer, G. Stampacchia: An Introduction to Variational Inequalities and Their Application, Academic Press, New York, 1980.Google Scholar
  9. B. Louro, J. F. Rodrigues: Remarks on the quasi-steady one-phase Stefan problem, Proc. of the Royal Soc. of Edinburgh 102A (1986) 263-275.https://doi.org/10.1017/S0308210500026354Google Scholar
  10. J. A. McGeough: Principles of Electrochemical Machining, Chapman and Hall, London, 1974.Google Scholar
  11. J. A. McGeough, H. Rasmussen: On the derivation of a quasi-stationary model in electrochemical machining, Journ. Inst. Math. Appl. (IMA) 13 (1974) 13-21.https://doi.org/10.1093/imamat/13.1.13Google Scholar
  12. U. Mosco: Convergence of convex sets and of solutions of variational inequalities, Advances in Mathematics 3 (1969) 510-585.https://doi.org/10.1016/0001-8708(69)90009-7Google Scholar
  13. J. F. Rodrigues: Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies 134, North-Holland Publishing Company, Amsterdam et al., 1987.Google Scholar
  14. J. F. Rodrigues: Variational methods in the Stefan problem, in: Phase Transitions and Hysteresis, A. Visintin (ed.), Lecture Notes in Mathematics 1584, Springer-Verlag, Berlin et al., 1994, 147-212.https://doi.org/10.1007/BFb0073397Google Scholar
  15. J. Steinbach: Evolutionary variational inequalities arising in non-isothermal Hele-Shaw flows, to appear in: Advances in Math. Sciences and Applications 8 (1998) 357-371.Google Scholar
  16. J. Steinbach: A generalized temperature-dependent, non-Newtonian Hele-Shaw flow in injection and compression moulding, Math. Methods in the Applied Sciences 20 (1997) 1199-1222.https://doi.org/10.1002/(SICI)1099-1476(19970925)20:14<1199::AID-MMA910>3.0.CO;2-5Google Scholar
  17. V. M. Volgin: Models for the evolution of the (shaping) boundary in the ECM process, in Russian, in: Proceedings of the All-Russian Conference 'Advanced Electrotechnology in Machine Building Industry', June 1997, Tula State University (Russia) (1997) 27-41.Google Scholar
  18. E. Wegert, D. Oestreich: On a non-symmetric problem in electrochemical maching, Math. Methods in the Applied Sciences 20 (1997) 841-854.https://doi.org/10.1002/(SICI)1099-1476(19970710)20:10<841::AID-MMA886>3.0.CO;2-LGoogle Scholar

Suggested citation

J. Steinbach. “On a Variational Inequality Containing a Memory Term with an Application in Electro-Chemical Machining.” Journal of Convex Analysis 5 (1998), No. 1, 63–80. https://doi.org/10.68381/jca05-6

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Published by Heldermann Verlag, 1998. Rights now held by Banach Press.