Variational Calculus with Elementary Convexity

1. Front Matter

   Pages i-xiv

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2. Review of Optimization in ℝ d

   1. Review of Optimization in ℝ^d

      • John L. Troutman

      Pages 1-9

3. Basic Theory

   1. Front Matter

      Pages 11-11

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   2. Standard Optimization Problems

      • John L. Troutman

      Pages 13-34

   3. Linear Spaces and Gâteaux Variations

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      Pages 35-51

   4. Minimization of Convex Functions

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      Pages 52-92

   5. The Lemmas of Lagrange and du Bois-Reymond

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      Pages 93-98

   6. Local Extrema in Normed Linear Spaces

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      Pages 99-141

   7. The Euler-Lagrange Equations

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      Pages 142-190

4. Advanced Topics

   1. Front Matter

      Pages 191-192

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   2. Piecewise C¹ Extremal Functions

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      Pages 193-229

   3. Variational Principles in Mechanics

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      Pages 230-270

   4. Sufficient Conditions for a Minimum

      • John L. Troutman

      Pages 271-325

5. Back Matter

   Pages 327-365

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The calculus of variations, whose origins can be traced to the works of Aristotle and Zenodoros, is now Ii vast repository supplying fundamental tools of exploration not only to the mathematician, but-as evidenced by current literature-also to those in most branches of science in which mathematics is applied. (Indeed, the macroscopic statements afforded by variational principles may provide the only valid mathematical formulation of many physical laws. ) As such, it retains the spirit of natural philosophy common to most mathematical investigations prior to this century. How­ ever, it is a discipline in which a single symbol (b) has at times been assigned almost mystical powers of operation and discernment, not readily subsumed into the formal structures of modern mathematics. And it is a field for which it is generally supposed that most questions motivating interest in the subject will probably not be answerable at the introductory level of their formulation. In earlier articles,1,2 it was shown through several examples that a complete characterization of the solution of optimization problems may be available by elementary methods, and it is the purpose of this work to explore further the convexity which underlay these individual successes in the context of a full introductory treatment of the theory of the variational calculus. The required convexity is that determined through Gateaux variations, which can be defined in any real linear space and which provide an unambiguous foundation for the theory.

• Convexity
• Konvexe Funktion
• Variationsrechnung
• calculus
• differential equation

• Department of Mathematics, Syracuse University, Syracuse, USA

  John L. Troutman

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