Książki

Nonlinear partial differential equations has become one of the main tools of mod ern mathematical analysis; in spite of seemingly contradictory terminology, the subject of nonlinear differential equations finds its origins in the theory of linear differential equations, and a large part of functional analysis derived its inspiration from the study of linear pdes. In recent years, several mathematicians have investigated nonlinear equations, particularly those of the second order, both linear and nonlinear and either in divergence or nondivergence form. Quasilinear and fully nonlinear differential equations are relevant classes of such equations and have been widely examined in the mathematical literature. In this work we present a new family of differential equations called "implicit partial differential equations", described in detail in the introduction (c.f. Chapter 1). It is a class of nonlinear equations that does not include the family of fully nonlinear elliptic pdes. We present a new functional analytic method based on the Baire category theorem for handling the existence of almost everywhere solutions of these implicit equations. The results have been obtained for the most part in recent years and have important applications to the calculus of variations, nonlin ear elasticity, problems of phase transitions and optimal design; some results have not been published elsewhere.

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.- 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.