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Implicit Partial Differential Equations

  • Bernard Dacorogna
  • Paolo Marcellini

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 37)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Introduction

    1. Bernard Dacorogna, Paolo Marcellini
      Pages 1-30
  3. First and Second Order Partial Differential Equations

    1. Front Matter
      Pages 31-31
    2. Bernard Dacorogna, Paolo Marcellini
      Pages 33-68
    3. Bernard Dacorogna, Paolo Marcellini
      Pages 69-93
    4. Bernard Dacorogna, Paolo Marcellini
      Pages 95-117
  4. Systems of Partial Differential Equations

    1. Front Matter
      Pages 119-119
    2. Bernard Dacorogna, Paolo Marcellini
      Pages 121-140
    3. Bernard Dacorogna, Paolo Marcellini
      Pages 141-165
  5. Applications

    1. Front Matter
      Pages 167-167
    2. Bernard Dacorogna, Paolo Marcellini
      Pages 169-203
    3. Bernard Dacorogna, Paolo Marcellini
      Pages 205-216
    4. Bernard Dacorogna, Paolo Marcellini
      Pages 217-222
  6. Appendix

    1. Front Matter
      Pages 223-223
    2. Bernard Dacorogna, Paolo Marcellini
      Pages 225-247
  7. Back Matter
    Pages 249-273

About this book

Introduction

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.

Keywords

Boundary value problem Lipschitz domain approximation property calculus calculus of variations nonlinear analysis odes/pdes partial differential equation

Authors and affiliations

  • Bernard Dacorogna
    • 1
  • Paolo Marcellini
    • 2
  1. 1.Department of MathematicsEcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Dipartimento di Matematica “U. Dini”Università di FirenzeFirenzeItaly

Bibliographic information

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