Implicit Partial Differential Equations

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. Front Matter....Pages i-xiii Introduction....Pages 1-30 Front Matter....Pages 31-31 First Order Equations....Pages 33-68 Second Order Equations....Pages 69-93 Comparison with Viscosity Solutions....Pages 95-117 Front Matter....Pages 119-119 Some Preliminary Results....Pages 121-140 Existence Theorems for Systems....Pages 141-165 Front Matter....Pages 167-167 The Singular Values Case....Pages 169-203 The Case of Potential Wells....Pages 205-216 The Complex Eikonal Equation....Pages 217-222 Front Matter....Pages 223-223 Appendix: Piecewise Approximations....Pages 225-247 Back Matter....Pages 249-273