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E-book: From Convexity to Nonconvexity

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This collection of papers is dedicated to the memory of Gaetano Fichera, a great mathematician and also a good friend to the editors. Regrettably it took an unusual amount of time to bring this collection out. This was primarily due to the fact that the main editor who had collected all of the materials, for this volume, P. D. Panagiotopoulos, died unexpectedly during the period when we were editing the manuscript. The other two editors in appreciation of Panagiotopoulos' contribution to this field, believe it is therefore fitting that this collection be dedicated to his memory also. The theme of the collection is centered around the seminal research of G. Fichera on the Signorini problem. Variants on this idea enter in different ways. For example, by bringing in friction the problem is no longer self-adjoint and the minimization formulation is not valid. A large portion of this collection is devoted to survey papers concerning hemivariational methods, with a main point of its application to nonsmooth mechanics. Hemivariational inequali­ ties, which are a generalization of variational inequalities, were pioneered by Panagiotopoulos. There are many applications of this theory to the study of non convex energy functionals occurring in many branches of mechanics. An area of concentration concerns contact problems, in particular, quasistatic and dynamic contact problems with friction and damage. Nonsmooth optimization methods which may be divided into the main groups of subgradient methods and bundle methods are also discussed in this collection.
Preface xi Frictional contact problems 1(14) Lars-Erick Andersson Anders Klarbring Introduction 1(1) Elementary example of non-uniqueness and non-existence 2(2) Classical formulation of the quasistatic frictional contact problem 4(1) The static problem 5(3) Steady sliding problem 8(1) Existence results for quasistatic friction problems 8(2) Conclusion 10(5) References 11(4) Solutions for quasilinear hemivariational inequalities 15(14) Siegfried Carl Introduction 15(1) Notations, hypotheses and the main result 16(4) Auxiliary results 20(4) Proof of the main result 24(5) Static problem 25(1) Example 25(1) Concluding remarks 26(1) References 27(2) A Survey on Nonsmooth Critical Point Theory 29(14) Marco Degiovanni Introduction 29(2) Critical point theory in metric spaces 31(2) Subdifferential calculus 33(2) Functionals of the calculus of variations 35(1) Functionals with quadratic dependence on the gradient 36(2) Area-type functionals 38(5) References 39(4) Exhaustive families of approximations revisited 43(8) V.F. Demyanov A.M. Rubinov Directional derivatives and generalizations 43(2) Exhaustive families of upper and lower approximations 45(6) References 49(2) Optimal shape design 51(16) Zdzislaw Denkowski Introduction 51(1) Preliminaries 52(1) State relations for physical systems 53(3) Abstract OSD and direct method 56(1) Mapping method and its applications 57(2) Some remarks on other methods for OSD problems 59(2) Relaxation in OSD problems 61(2) Lower semicontinuity of functionals in OSD problems 63(4) References 63(4) Duality in Nonconvex Finite Deformation Theory 67(18) David Yang Gao Introduction 67(2) Framework and Abstract Boundary Value Problem 69(2) Conjugate Stress-Strain Tensors and Gap Functions 71(3) Potential Extremum Principle 74(1) Classical Complementary Energy Principles 75(2) Generalized Variational Principles and Triality Theory 77(1) Pure Complementary Energy Principles and Minimax Theory 78(7) References 80(5) Contact Problems in Multibody Dynamics 85(26) Friedrich Pfeiffer Christoph Glocker Introduction 85(1) The Evolution of a Theory 86(13) Present Mathematical Formulation 99(8) Conclusions 107(4) References 107(4) Hyperbolic Hemivariational Inequality 111(12) D. Goeleven1 D. Motreanu2 Introduction and formulation of nonsmooth hyperbolic problem 111(2) Finite dimensional approximation 113(4) Main Results 117(6) References 121(2) Time-integration algorithms 123(14) Klaus Hackl Introduction 123(1) An augmented principle of maximum plastic dissipation 124(2) The Evolution Problem 126(1) Time-Integration Algorithms and their Stability Properties 126(4) Algorithms Involving Operator-Split 130(5) Conclusion 135(2) References 135(2) Contact Stress Optimization 137(10) J. Haslinger Introduction 137(1) Formulation of the problem 138(9) References 145(2) Recent results in contact problems with Coulomb friction 147(14) J. Jarusek C. Eck Introduction 147(1) The static case 148(4) Dynamic problem with contact condition in displacement and given friction 152(3) Dynamic problem with Coulomb friction and contact condition in velocities 155(2) Appendix: Thermal aspects of friction 157(2) Conclusion 159(2) References 159(2) Polarization fields in linear piezoelectricity 161(16) P. Bisegna F. Maceri Introduction 161(1) The linear piezoelectric problem 162(3) Weak formulations of the linear piezoelectric problem 165(1) Hashin-Shtrikman type variational principles 166(4) Conclusions 170(7) References 170(7) Survey of the methods for nonsmooth optimization 177(16) M. M. Makela Introduction 177(1) Convex optimization 178(7) Nonconvex optimization 185(8) References 188(5) Hemivariational inequalities and hysteresis 193(14) M. Miettinen References 205(2) Non convex aspects of dynamics with impact 207(16) L. Paoli M. Schatzmann References 221(2) On Global Properties of D.C. Functions 223(10) L. N. Polyakova References 231(2) Variational-Hemivariational Inequalities 233(10) G. Dinca G. Pop References 241(2) Perturbations of Eigenvalue Problems 243(12) Vicentiu D. Radulescu Implicit variational inequalities arising in frictional unilateral contact mechanics: analysis and numerical solution of quasistatic problems 255(14) Marius Cocu Michel Raous Introduction 255(1) Quasistatic contact problems with friction 256(3) Extension to a model coupling adhesion and friction 259(4) Numerical methods 263(6) References 265(4) Regularity for variational inequalities 269(14) Rainer Schumann Introduction 269(1) Variational inequalities and their applications 270(2) Regularity 272(11) References 280(3) A Survey of 1-D Problems of Dynamic Contact and Damage 283(14) Meir Shillor Introduction 283(1) Preliminaries 284(1) Dynamic Thermoviscoelastic Contact of a Rod 285(2) Vibrations of a Beam between Two Stops 287(3) A Beam in Frictional Contact 290(2) The Elastic Rod with Damage 292(5) References 294(3) Nonconvexity in plasticity and damage 297(14) Georgios E. Stavroulakis Introduction 297(1) Nonsmooth modeling in mechanics 298(3) Elastoplasticity 301(4) Damage mechanics 305(6) References 307(4) Augmented Lagrangian Methods for Contact Problems 311(22) Jozef Joachim Telega Andrzej Galka Introduction 311(1) General results 312(2) Ito and Kunisch augmented Lagrangian methods 314(4) Contact Problems 318(4) Parameter Estimation and Optimal Control 322(4) Image Restoration 326(7) References 328(5) Mountain Pass Theorems 333(12) Stepan A. Tersian Introduction 333(1) Deformation theorems and (PS) conditions 334(5) Mountain pass theorems 339(6) References 343(2) Proximal Methods for Variational Inequalities with Set-Valued Monotone Operators 345(18) A. Kaplan R. Tichatschke Introduction 345(3) Multi-step Proximal Regularization Scheme 348(1) Convergence Analysis 349(14) References 358(5) Simons Problem 363(18) Miroslaw Przeworski Dariusz Zagrodny Introduction 363(2) Some basic facts and definitions 365(3) A solution of the generalized Simons problem 368(4) Locating elements of graphs of maximal monotone operators 372(9) References 378(3) Density estimates of Blake & Zisserman functional 381(10) Michele Carriero Antonio Leaci Franco Tomarelli Introduction 381(2) Notation and preliminary results 383(3) Local weak minimizers and essential minimizing triplets. 386(1) Density estimates for essential minimizing triplets. 387(3) A counterexample 390(1) References 391
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