Direct Methods in the Calculus of Variations - Second Edition

In this review of Direct Methods in the Calculus of Variations readers will find a thorough, reference-grade treatment best suited to advanced students and researchers. The second edition builds substantially on the original by adding nearly half new material and introducing quasiconvex analysis, making it the single biggest reason to consult this book: it updates a classic with modern, research-relevant topics. The tone is scholarly rather than introductory, so the work serves as both a course text for advanced classes and a long-term reference for those working in the calculus of variations.

Key Features

• Expanded content: Almost half the book is new material, offering readers recent developments and updated perspectives compared with the out-of-print first edition.
• Quasiconvex analysis focus: The book now contains a substantial treatment of quasiconvex analysis, providing a modern toolkit for variational problems where quasiconvexity is central.
• Reference-oriented structure: Organized as a reference, the text is suitable for researchers who need precise statements and proofs rather than survey-level coverage.
• Course usability: Despite its reference depth, the book remains usable for an advanced course in the calculus of variations, with material arranged to support teaching and study.
• Continuity with earlier edition: The core and structure mirror the first edition, easing transition for readers familiar with the earlier work while extending it substantially.

Who It's For

The primary audience is graduate students, doctoral researchers, and established mathematicians working in analysis, partial differential equations, and the calculus of variations who need a rigorous, up-to-date reference. It is particularly valuable for those studying variational methods where quasiconvexity plays a role and for instructors designing an advanced course.

Readers seeking an introductory textbook or a gently paced introduction to variational ideas should look elsewhere; this edition assumes a mature mathematical background and aims at depth and completeness rather than elementary exposition.

Pros & Cons

Pros

• Comprehensive update: contains nearly half new material that reflects progress in the field over fifteen years.
• Dedicated quasiconvex analysis coverage that fills a gap in many earlier texts.
• Structured for use as both a course text and a long-term reference for researchers.

Cons

• Advanced level and dense presentation make it less suitable for beginners or casual readers.

Specifications

Title	Direct Methods in the Calculus of Variations (Second Edition)
Series	Applied Mathematical Sciences, 78
Author	Bernard Dacorogna
Edition	Second edition
Scope	Reference and advanced course text emphasizing quasiconvex analysis
Audience	Graduate students, researchers, instructors in mathematics

Our Verdict

Direct Methods in the Calculus of Variations, second edition, is a worthwhile acquisition for anyone needing a rigorous, current reference on variational methods and quasiconvex analysis. It represents good value for advanced students and researchers because it updates an important earlier text with substantial new material while preserving the original structure for classroom and reference use.

Frequently Asked Questions

Is this book suitable for a graduate course?
Yes; the book is explicitly usable for an advanced course, though it assumes a solid background in analysis.

Does the second edition add new topics?
Yes; nearly half the content is new and it notably develops the topic of quasiconvex analysis.

Who wrote the book?
The author is Bernard Dacorogna, and the work continues the earlier edition now out of print.