Calculus of Variations and Partial Differential Equations

1. Front Matter

   Pages I-IX

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2. Geometric Evolution Problems

   1. Front Matter

      Pages 1-1

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   2. Introduction to Part I

      • Luigi Ambrosio, Norman Dancer, Giuseppe Buttazzo, Antonio Marino, M. K. V. Murthy

      Pages 3-4

   3. Geometric evolution problems, distance function and viscosity solutions

      • L. Ambrosio

      Pages 5-93

   4. Variational models for phase transitions, an approach via Γ-convergence

      • G. Alberti

      Pages 95-114

   5. Some aspects of De Giorgi’s barriers for geometric evolutions

      • G. Bellettini, M. Novaga

      Pages 115-151

   6. Partial Regularity for Minimizers of Free Discontinuity Problems with p-th Growth

      • A. Leaci

      Pages 153-169

   7. Free discontinuity problems and their non-local approximation

      • A. Braides

      Pages 171-180

3. Degree Theory on Convex Sets and Applications to Bifurcation

   1. Front Matter

      Pages 181-181

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   2. Introduction to Part II

      • Luigi Ambrosio, Norman Dancer, Giuseppe Buttazzo, Antonio Marino, M. K. V. Murthy

      Pages 183-184

   3. Degree theory on convex sets and applications to bifurcation

      • E. N. Dancer

      Pages 185-225

   4. Nonlinear elliptic equations involving critical Sobolev exponents

      • D. Passaseo

      Pages 227-241

   5. On the existence and multiplicity of positive solutions for semilinear mixed and Neumann elliptic problems

      • G. Cerami

      Pages 243-258

   6. Solitons and Relativistic Dynamics

      • V. Benci, D. Fortunato

      Pages 259-283

   7. An algebraic approach to nonstandard analysis

      • V. Benci

      Pages 285-326

4. Back Matter

   Pages 327-347

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The link between Calculus of Variations and Partial Differential Equations has always been strong, because variational problems produce, via their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can often be studied by variational methods. At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on a classical topic (the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to pde's resp.), in a self-contained presentation accessible to PhD students, bridging the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and nicely illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.

• Calculus of Variations
• Lagrangian mechanics
• direct variational methods
• geometric measure theory
• partial differential equation
• partial differential equations
• topological variational methods

• Dipartimento di Matematica, Università di Pisa, Pisa, Italy

  Giuseppe Buttazzo, Antonio Marino, M. K. V. Murthy

• Scuola Normale Superiore, Pisa, Italy

  Luigi Ambrosio

• School of Mathematics, University of Sydney, Sydney, Australia

  Norman Dancer

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