关于《Modern Methods in the Calculus of Variations》第二卷的详情咨询

Hey there! Let me break down everything you need to know about the second volume in Irene Fonseca and Giovanni Leoni's calculus of variations series, building on the info you shared about the first book.

First, a quick recap of the first volume to set the stage: Modern Methods in the Calculus of Variations: $L^p$ Spaces is the first of two books focused on modern variational techniques. As its "About this book" section states:

This is the first of two books on methods and techniques in the calculus of variations. Contemporary arguments are used throughout the text to streamline and present in a unified way classical results, and to provide novel contributions at the forefront of the theory.

This book addresses fundamental questions related to lower semicontinuity and relaxation of functionals within the unconstrained setting, mainly in $L^p$ spaces. It prepares the ground for the second volume where the variational treatment of functionals involving fields and their derivatives will be undertaken within the framework of Sobolev spaces.

And as the reviews highlight, it's a highly regarded, self-contained resource for both students and researchers:

"This book is the first of two volumes in the calculus of variations and measure theory. The main objective of this book is to introduce necessary and sufficient conditions for sequential lower semicontinuity of functionals on Lp-spaces. … This book is very nicely written, self-contained and it is an excellent and modern introduction to the calculus of variations." (Jean-Pierre Raymond, Zentrablatt MATH, Vol. 1153, 2009)

“This is the first of a two-volume introduction into direct methods in the calculus of variations. Its main topic is the analysis of necessary and sufficient conditions for lower semicontinuity on Lp-spaces, as well as of relaxation techniques. … The book provides a well-written and self-contained introduction to an active area of research and will be valuable both to graduate students as an introduction and to researchers in the field as a reference work.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)

Now, onto the second volume:

• Title: Modern Methods in the Calculus of Variations: Sobolev Spaces (authored by the same pair, Irene Fonseca and Giovanni Leoni)
• Core Focus: As teased in the first volume, this book centers on variational problems involving functionals that depend on fields and their derivatives, all within the Sobolev space framework. It picks up exactly where the first volume left off, extending the modern, unified approach to this more complex setting.
• Audience & Value: Just like its predecessor, it's self-contained, rigorously written, and caters to both graduate students looking to deepen their understanding of variational methods and researchers needing a reliable reference for Sobolev-space-based variational theory.

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