Implicit Partial Differential Equations: 37 - Rilegato

Dacorogna, Bernard; Marcellini, Paolo

Rilegato

ISBN 10: 0817641211 ISBN 13: 9780817641214

Casa editrice: Springer Basel AG, 1999

Riassunto

Nonlinear partial differential equations has become one of the main tools of mod­ ern mathematical analysis;

Su questo libro

Recensione

"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

Contenuti

1 Introduction.- 1.1 The first order case.- 1.1.1 Statement of the problem.- 1.1.2 The scalar case.- 1.1.3 Some examples in the vectorial case.- 1.1.4 Convexity conditions in the vectorial case.- 1.1.5 Some typical existence theorems in the vectorial case.- 1.2 Second and higher order cases.- 1.2.1 Dirichlet-Neumann boundary value problem.- 1.2.2 Fully nonlinear partial differential equations.- 1.2.3 Singular values.- 1.2.4 Some extensions.- 1.3 Different methods.- 1.3.1 Viscosity solutions.- 1.3.2 Convex integration.- 1.3.3 The Baire category method.- 1.4 Applications to the calculus of variations.- 1.4.1 Some bibliographical notes.- 1.4.2 The variational problem.- 1.4.3 The scalar case.- 1.4.4 Application to optimal design in the vector-valued case.- 1.5 Some unsolved problems.- 1.5.1 Selection criterion.- 1.5.2 Measurable Hamiltonians.- 1.5.3 Lipschitz boundary data.- 1.5.4 Approximation of Lipschitz functions by smooth functions.- 1.5.5 Extension of Lipschitz functions and compatibility conditions.- 1.5.6 Existence under quasiconvexity assumption.- 1.5.7 Problems with constraints.- 1.5.8 Potential wells.- 1.5.9 Calculus of variations.- I First Order Equations.- 2 First and Second Order PDE’s.- 2.1 Introduction.- 2.2 The convex case.- 2.2.1 The main theorem.- 2.2.2 An approximation lemma.- 2.2.3 The case independent of (x, u).- 2.2.4 Proof of the main theorem.- 2.3 The nonconvex case.- 2.3.1 The pyramidal construction.- 2.3.2 The general case.- 2.4 The compatibility condition.- 2.5 An attainment result.- 3 Second Order Equations.- 3.1 Introduction.- 3.2 The convex case.- 3.2.1 Statement of the result and some examples.- 3.2.2 The approximation lemma.- 3.2.3 The case independent of lower order terms.- 3.2.4 Proof of the main theorem.- 3.3 Some extensions.- 3.3.1 Systems of convex functions.- 3.3.2 A problem with constraint on the determinant.- 3.3.3 Application to optimal design.- 4 Comparison with Viscosity Solutions.- 4.1 Introduction.- 4.2 Definition and examples.- 4.3 Geometric restrictions.- 4.3.1 Main results.- 4.3.2 Proof of the main results.- 4.4 Appendix.- 4.4.1 Subgradient and differentiability of convex functions.- 4.4.2 Gauges and their polars.- 4.4.3 Extension of Lipschitz functions.- 4.4.4 A property of the sub and super differentials.- II Systems of Partial Differential Equations.- 5 Some Preliminary Results.- 5.1 Introduction.- 5.2 Different notions of convexity.- 5.2.1 Definitions and basic properties (first order case).- 5.2.2 Definitions and basic properties (higher order case).- 5.2.3 Different envelopes.- 5.3 Weak lower semicontinuity.- 5.3.1 The first order case.- 5.3.2 The higher order case.- 5.4 Different notions of convexity for sets.- 5.4.1 Definitions.- 5.4.2 The different convex hulls.- 5.4.3 Further properties of rank one convex hulls.- 5.4.4 Extreme points.- 6 Existence Theorems for Systems.- 6.1 Introduction.- 6.2 An abstract result.- 6.2.1 The relaxation property.- 6.2.2 Weakly extreme sets.- 6.3 The key approximation lemma.- 6.4 Sufficient conditions for the relaxation property.- 6.4.1 One quasiconvex equation.- 6.4.2 The approximation property.- 6.4.3 Relaxation property for general sets.- 6.5 The main theorems.- III Applications.- 7 The Singular Values Case.- 7.1 Introduction.- 7.2 Singular values and functions of singular values.- 7.2.1 Singular values.- 7.2.2 Functions depending on singular values.- 7.2.3 Rank one convexity in dimension two.- 7.3 Convex and rank one convex hulls.- 7.3.1 The case of equality.- 7.3.2 The main theorem for general matrices.- 7.3.3 The diagonal case in dimension two.- 7.3.4 The symmetric case in dimension two.- 7.4 Existence of solutions (the first order case).- 7.5 Existence of solutions (the second order case).- 8 The Case of Potential Wells.- 8.1 Introduction.- 8.2 The rank one convex hull.- 8.3 Existence of solutions.- 9 The Complex Eikonal Equation.- 9.1 Introduction.- 9.2 The convex and rank one convex hulls.- 9.3 Existence of solutions.- IV Appendix.- 10 Appendix: Piecewise Approximations.- 10.1 Vitali covering theorems and applications.- 10.1.1 Vitali covering theorems.- 10.1.2 Piecewise affine approximation.- 10.2 Piecewise polynomial approximation.- 10.2.1 Approximation of functions of class CN.- 10.2.2 Approximation of functions of class WN,?.- References.

Dettagli dell'edizione

• Editore: Springer Basel AG
• Data di pubblicazione: 1999
• Lingua: Inglese
• ISBN 10: 0817641211
• ISBN 13: 9780817641214
• Rilegatura: Copertina rigida
• Numero di pagine: 292
• Contatto del produttore: ProductSafety@springernature.com; ProductSafety@springernature.com; ; Tiergartenstr. 17; Heidelberg; 69121; Germania