Implicit Partial Differential Equations

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