Implicit Partial Differential Equations - Bernard Dacorogna

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Sommaire:
1 Introduction.- 1.1 The first order case.- 1.2 Second and higher order cases.- 1.3 Different methods.- 1.4 Applications to the calculus of variations.- 1.5 Some unsolved problems.- I First Order Equations.- 2 First and Second Order PDE's.- 3 Second Order Equations.- 4 Comparison with Viscosity Solutions.- II Systems of Partial Differential Equations.- 5 Some Preliminary Results.- 6 Existence Theorems for Systems.- III Applications.- 7 The Singular Values Case.- 8 The Case of Potential Wells.- 9 The Complex Eikonal Equation.- IV Appendix.- 10 Appendix: Piecewise Approximations.- References.

Provides a self-contained development of the new kind of differential equations... Includes many examples helpful in understanding the theory and is well [and] clearly written. -Mathematical Reviews ...Topics of the book have important applications in variational calculus, nonlinear elasticity, phase transitions, and optimal design. In particular, Part IV (Chapters 7-9) includes the singular value case, the case of potential wells, and the complex eikonal equation. A lot of background material, as viscosity solutions, quasiconvexity and related topics, piecewise polynomial approximation in Sobolev spaces, convex integration, or nonclassical Vitali-type covering theorems, is presented in the book, which makes it essentially self-contained. Most of the results are original from the authors, some results are published even first time. This makes the book a unique introduction to this new topic, and will be found useful by experts in nonlinear analysis in general and PDEs in particular, as well as by advanced graduate students in this field. -Zentralblatt Math