Abstract
Optimal shape design problems for systems governed by an elliptic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.
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J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
L. Boccardo and F. Murat, Nouveaux résults de convergence dans des problemes unilatéraux, in Nonlinear PDEs and Their Appl., Collège de France Seminar, Vol. II, Research Notes in Math., 60, Pitman, London, 1982, 64–85.
H. Brezis, Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, North Holland, Amsterdam, 1973.
G. Buttazzo and G. Dal Maso, An existence result for a class of shape optimization problems, Preprint 124/M, S. I. S. S. A., 1992.
K. C. Chang, Variational methods for nondifferentiable functionals and applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102–129.
F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.
G. Dal Maso, Some necessary and sufficient conditions for the convergence of sequences of unilateral convex sets, J. Funct. Anal., 62 (1985), 119–159.
M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity, Nonlinear Anal., Theory, Methods and Appl., 9 (1985), 1401–1443.
Z. Denkowski and S. Migórski, Optimal shape design for hemivariational inequalities, Proc. Conference on Topological Methods in Differential Equations and Dynamical Systems, Kraków, Poland, July 17–20, 1996, Universitatis Iagellonicae Acta Math., 1997, in press.
Z. Denkowski and S. Migórski, Optimal shape design for elliptic hemivariational inequalities in nonlinear elasticity, Proc. 12th Conference on Variational Calculus, Optimal Control and Applications, Trassenheide, Germany, September 23–27, 1996, Birkhäuser-Verlag, 1997, in press.
G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.
J. Haslinger and P. Neittaanmäki, Finite Element Approximation of Optimal Shape Design. Theory and Applications, Wiley, New York, 1988.
J. Haslinger, P. Neittaanmäki and T. Tiihonen, Shape optimization in contact problems based on penalization of the state inequality, Aplikace Mat., 31 (1986), 54–77.
J. Haslinger and P. D. Panagiotopoulos, Optimal control of hemivariational inequalities, Lecture Notes in Control and Info. Sci., 125, J. Simon (Ed.), 128–139, Springer-Verlag, 1989.
J. Haslinger and P. D. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Anal. Theory Methods Appl., 24 (1995), 105–119.
W. B. Liu and J. E. Rubio, Optimal shape design for systems governed by variational inequalities, Part 1: Existence theory for elliptic case, Part 2: Existence theory for evolution case, J. Optim. Th. Appl., 69 (1991), 351–371, 373–396.
M. Miettinen, Approximation of Hemivariational Inequalities and Optimal Control Problems, Thesis, University of Jyväskylä, Finland, Report 59, 1993.
M. Miettinen, M. M. Mäkeläand J. Haslinger, On numerical solution of hemivariational inequalities by nonsmooth optimization methods, J. Global Optim., 6 (1995), 401–425.
J. J. Moreau, Sur les lois de frottement, de plasticité et de viscosité, C. R. Acad. Sci. Paris, 271A (1970), 608–611.
F. Murat and J. Simon, Sur le Controle par un Domaine Geometrique, Preprint no. 76015, University of Paris 6, 1976.
F. Murat and J. Simon, Etude de problemes d'optimal design, Proc. 7th IFIP Conference, Optimization techniques: Modelling and Optimization in the Service of Man, Nice, September 1972, Lect. Notes in Computer Sci., 41, 54–62, Springer-Verlag, 1976.
Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities, Dekker, New York, 1995.
P. Neittaanmäki, On the control of the domain in variational inequalities, in: Differential Equations and Control Theory, V. Barbu (Ed.), Pitman Research Notes in Math., 250 (1991), 228–247.
P. D. Panagiotopoulos, Nonconvex problems of semipermeable media and related topics, Z. Angew. Math. Mech., 65 (1985), 29–36.
P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, Basel, 1985.
P. D. Panagiotopoulos, Coercive and semicoercive hemivariational inequalities, Nonlinear Anal. Theory Methods Appl., 16 (1991), 209–231.
P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993.
P. D. Panagiotopoulos, Hemivariational inequality and fan-variational inequalities. New applications and results, Atti Sem. Mat. Fis. Univ. Modena, XLIII (1995), 159–191.
O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, 1984.
J. Rauch, Discontinuous semilinear differential equations and multiple valued maps, Proc. Amer. Math. Soc., 64 (1977), 277–282.
K. Salmenjoki, On numerical methods for shape design problems, Thesis, University of Jyväskylä, Finland, 1991.
J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139–167.
T. Tiihonen, Abstract approach to a shape design problem for variational inequalities, Preprint 62, University of Jyväskylä, Finland, 1987.
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Denkowski, Z., Migórski, S. Optimal Shape Design Problems for a Class of Systems Described by Hemivariational Inequalities. Journal of Global Optimization 12, 37–59 (1998). https://doi.org/10.1023/A:1008299801203
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DOI: https://doi.org/10.1023/A:1008299801203