Dissipative and Non-Dissipative Evolutionary Quasi-Variational Inequalities with Gradient Constraints

  • M. Hintermüller
  • C. N. RautenbergEmail author
  • N. Strogies


Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.


Quasi-variational inequality Gradient constraint Dissipative and non-dissipative processes Variable splitting solver 

Mathematics Subject Classification (2010)

35K86 47J20 49J40 49M15 65J15 65K10 


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  1. 1.
    Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems birkhäuser (2010)Google Scholar
  2. 2.
    Azevedo, A., Miranda, F., Santos, L.: Variational and quasivariational inequalities with first order constraints. J. Math. Anal. Appl. 397(2), 738–756 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Azevedo, A., Miranda, F., Santos, L.: Stationary Quasivariational Inequalities with Gradient Constraint and Nonhomogeneous Boundary Conditions, pp. 95–112. Springer, Berlin (2014)zbMATHGoogle Scholar
  4. 4.
    Baiocchi, C., Capelo, A.: Variational and quasivariational inequalities Wiley-Interscience (1984)Google Scholar
  5. 5.
    Barret, J.W., Prigozhin, L.: A quasi-variational inequality problem in superconductivity. Mathematical Models and Methods in Applied Sciences 20(5), 679–706 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barrett, J.W., Prigozhin, L.: A quasi-variational inequality problem arising in the modeling of growing sandpiles. ESAIM Math. Model. Numer. Anal. 47(4), 1133–1165 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011). With a foreword by Hédy AttouchGoogle Scholar
  8. 8.
    Bensoussan, A., Lions, J.L.: Controle impulsionnel et inéquations quasi-variationnelles d’évolutions. C. R. Acad. Sci. Paris 276, 1333–1338 (1974)zbMATHGoogle Scholar
  9. 9.
    Beremlijski, P., Haslinger, J., Kocvara, M., Outrata, J.: Shape optimization in contact problems with Coulomb friction. SIAM J. Optim. 13, 561–587 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brézis, H., Sibony, M.: Equivalence de deux inéquations variationnelles et applications. Archive Rat. Mech. Anal. 41, 254–265 (1971)CrossRefzbMATHGoogle Scholar
  11. 11.
    Brézis, H., Stampacchia, G.: Sur la régularité de la solution d’inéquations elliptiques. Bulletin de la S. M. F. 96, 153–180 (1968)zbMATHGoogle Scholar
  12. 12.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, Berlin (2008)zbMATHGoogle Scholar
  13. 13.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: Introduction to nonlinear analysis: Applications Kluwer (2003)Google Scholar
  14. 14.
    DiBenedetto, E.: Real analysis. Advanced Texts Series. Birkhauser, Cambridge (2002). Google Scholar
  15. 15.
    Duvaut, G., Lions, J.P.: Les inéquations en mécanique et en physique. Dunod, Paris (1972)zbMATHGoogle Scholar
  16. 16.
    Fattorini, H.O.: Infinite dimensional optimization and control theory. Cambridge university press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  17. 17.
    Fukao, T., Kenmochi, N.: Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs. Math. Bohem. 139(2), 391–399 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fukao, T., Kenmochi, N.: Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete Contin. Dyn. Syst. 35(6), 2523–2538 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Glowinski, R., Lions, J.P., Trémolières, R.: Numerical analysis of variational inequalities North-Holland (1981)Google Scholar
  20. 20.
    Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups American Mathematical Society (1957)Google Scholar
  22. 22.
    Hintermüller, M., Kopacka, I.: Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm. SIAM J. Optim. 20(2), 868–902 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hintermüller, M., Rasch, J.: Several path-following methods for a class of gradient constrained variational inequalities. Comput. Math. Appl. 69(10), 1045–1067 (2015). MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hintermüller, M., Rautenberg, C.N.: A sequential minimization technique for elliptic quasi-variational inequalities with gradient constraints. SIAM J. Optim. 22(4), 1224–1257 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hintermüller, M., Rautenberg, C.N.: Parabolic quasi-variational inequalities with gradient-type constraints. SIAM J. Optim. 23(4), 2090–2123 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hintermüller, M., Rautenberg, C.N.: On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities. Port. Math. 74(1), 1–35 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hintermüller, M., Rautenberg, C.N., Rösel, S.: Density of convex intersections and applications. In: Proceedings of the Royal Society of London A: Mathematical Physical and Engineering Sciences 473(2205) (2017)Google Scholar
  28. 28.
    Kadoya, A., Kenmochi, N., Niezgódka, M.: Quasi-variational inequalities in economic growth models with technological development. Adv. Math. Sci. Appl. 24(1), 185–214 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kano, R., Kenmochi, N., Murase, Y.: Parabolic quasi-variational inequalities with non-local constraints. Adv. Math. Sci. Appl. 19(2), 565–583 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Kano, R., Murase, Y., Kenmochi, N.: Nonlinear evolution equations generated by subdifferentials with nonlocal constraints. In: Nonlocal and abstract parabolic equations and their applications, Banach Center Publ., vol. 86, pp. 175–194. Polish Acad. Sci. Inst. Math., Warsaw (2009).
  31. 31.
    Kenmochi, N.: Parabolic quasi-variational diffusion problems with gradient constraints. Discrete Contin. Dyn. Syst. Ser. S 6(2), 423–438 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. SIAM (2000)Google Scholar
  33. 33.
    Kleinhans, M.G., Markies, H., de Vet, S.J., in ’t Veld, A.C., Postema, F.N.: Static and dynamic angles of repose in loose granular materials under reduced gravity. Journal of Geophysical Research: Planets 116(E11), n/a–n/a (2011). CrossRefGoogle Scholar
  34. 34.
    Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-variational Inequalities in Mechanics. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  35. 35.
    Kunze, M., Rodrigues, J.: An elliptic quasi-variational inequality with gradient constraints and some of its applications. Mathematical Methods in the Applied Sciences 23, 897–908 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lions, J.L.: Sur le côntrole optimal des systemes distribuées. Enseigne 19, 125–166 (1973)zbMATHGoogle Scholar
  37. 37.
    Lions, J.L.: Asymptotic behaviour of solutions of variational inequalitites with highly oscillating coefficients. Applications of Methods of Functional Analysis to Problems in Mechanics. In: Proceedings of the Joint Symposium IUTAM/IMU. Lecture Notes in Mathematics, p 503. Springer, Berlin (1975)Google Scholar
  38. 38.
    Miranda, F., Rodrigues, J.F., Santos, L.: A class of stationary nonlinear maxwell systems. Mathematical Models and Methods in Applied Sciences 19(10), 1883–1905 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Miranda, F., Rodrigues, J.F., Santos, L.: On a p-curl system arising in electromagnetism. Discrete Contin. Dyn. Syst. Ser. S 5(3), 605–629 (2012)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Mosco, U.: Convergence of convex setis and solutions of variational inequalities. Adv. Math. 3(4), 510–585 (1969)CrossRefzbMATHGoogle Scholar
  41. 41.
    Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 3, 373–375 (2009)CrossRefzbMATHGoogle Scholar
  42. 42.
    Prigozhin, L.: Quasivariational inequality describing the shape of a poured pile. Zhurnal Vichislitel’noy Matematiki i Matematicheskoy Fiziki 7, 1072–1080 (1986)MathSciNetGoogle Scholar
  43. 43.
    Prigozhin, L.: Sandpiles and river networks: extended systems with non-local interactions. Phys. Rev. E 49, 1161–1167 (1994)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Prigozhin, L.: On the Bean critical-state model in superconductivity. Eur. J. Appl. Math. 7, 237–247 (1996)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Prigozhin, L.: Sandpiles, river networks, and type-ii superconductors. Free Boundary Problems News 10, 2–4 (1996)Google Scholar
  46. 46.
    Prigozhin, L.: Variational model of sandpile growth. Euro. J. Appl. Math. 7, 225–236 (1996)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Rodrigues, J.F.: Obstacle problems in mathematical physics North-Holland (1987)Google Scholar
  48. 48.
    Rodrigues, J.F., Santos, L.: A parabolic quasi-variational inequality arising in a superconductivity model. Ann. Scuola Norm. Sup. Pisa Cl. Sci. XXIX, 153–169 (2000)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Rodrigues, J.F., Santos, L.: Quasivariational solutions for first order quasilinear equations with gradient constraint. Arch. Ration. Mech. Anal. 205(2), 493–514 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Showalter, R.E.: Monotone operators in banach space and nonlinear partial differential equations american mathematical society (1997)Google Scholar
  51. 51.
    Simon, J.: Compact sets in the space L p(0,T; B). Annali di Matematica pure ed applicata CXLVI(IV), 65–96 (1987)zbMATHGoogle Scholar
  52. 52.
    Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  53. 53.
    Verfürth, R.: A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). CrossRefzbMATHGoogle Scholar
  54. 54.
    Willett, D., Wong, J.: On the discrete analogues of some generalizations of Gronwall’s inequality. Monatshefte für Mathematik 69(4), 362–367 (1965). MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Department of MathematicsHumboldt-University of BerlinBerlinGermany

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