Elements of Functional Analysis: From Banach Spaces to Elliptic Problems
暫譯: 函數分析要素：從巴拿赫空間到橢圓問題

Mazón, José M.

Elements of Functional Analysis: From Banach Spaces to Elliptic Problems
  • 出版商: Springer
  • 出版日期: 2026-08-03
  • 售價: $2,530
  • 貴賓價: 9.5 折 $2,403
  • 語言: 英文
  • 頁數: 442
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3032292093
  • ISBN-13: 9783032292094
  • 相關分類: 工程數學 Engineering-mathematics

    • 海外代購書籍(需單獨結帳)

商品描述

Functional analysis can be understood as a shift, within mathematical analysis, from the study of individual functions to the study of function spaces, their structures, and the mappings between them. Developed primarily during the 20th century, this theory continues to be essential for the study of partial differential equations.

This textbook begins with the general theory: Banach spaces, Hilbert spaces, and the spectral theory of operators. It then proceeds to a thorough account of Schwartz's Theory of Distributions and some of its applications, such as the fundamental solutions of classical operators in physics, the Malgrange-Ehrenpreis Theorem, hypoelliptic operators, and the Schrödinger equation. An extensive chapter is dedicated to the study of Sobolev spaces, including functions of bounded variation. These spaces provide the appropriate framework for the study of elliptic boundary value problems, which form the focus of the final chapter.

Drawing on years of teaching experience, this textbook is ideal for introductory graduate courses, featuring historical notes and numerous exercises that reinforce learning and complement the core material.

商品描述(中文翻譯)

功能分析可以理解為數學分析中的一個轉變，從對單個函數的研究轉向對函數空間、其結構及其之間映射的研究。這一理論主要在20世紀發展起來，至今仍然對偏微分方程的研究至關重要。

本教科書首先介紹一般理論：巴拿赫空間（Banach spaces）、希爾伯特空間（Hilbert spaces）以及算子的譜理論（spectral theory of operators）。接著，書中詳細闡述了施瓦茨分佈理論（Schwartz's Theory of Distributions）及其一些應用，例如物理學中經典算子的基本解、馬爾格朗日-艾倫普賽斯定理（Malgrange-Ehrenpreis Theorem）、假橢圓算子（hypoelliptic operators）以及薛丁格方程（Schrödinger equation）。有一章專門致力於索博列夫空間（Sobolev spaces）的研究，包括有界變差函數（functions of bounded variation）。這些空間為研究橢圓邊值問題提供了適當的框架，這也是最後一章的重點。

本教科書基於多年教學經驗，特別適合用於研究生的入門課程，並包含歷史註解和大量練習題，以加強學習並補充核心材料。

作者簡介

José M. Mazón is an emertius professor of mathematical analysis at the University of Valencia. He is a specialist in nonlinear partial differential equations, a field in which he has published more than 160 research articles in prestigious journals. His research topics include singular and degenerate nonlinear PDEs, image processing equations, flux limited diffusion equations, mass transport theory and nonlocal equations. Some of his results have been published in six monographs: Parabolic Quasilinear Equations Minimizing Linear Growth Functional (Birkhäuser, winner of the 2023 Ferran Sunyer i Balaguer Prize), Nonlocal Diffusion Problems (American Mathematical Society, 2010), Nonlocal Perimeter, Curvature and Minimal Surfaces (Birkhäuser, 2019), Variational and Diffusion Problems in Random Walk Spaces (Birkhäuser, 2023), Functions of Least Gradient (Birkhäuser, 2024) and Weak Solutions to Gradient Flows in Metric Measure Spaces (Cambridge University Press, 2026).

作者簡介(中文翻譯)

José M. Mazón 是瓦倫西亞大學的榮譽教授，專攻數學分析。他是非線性偏微分方程的專家，在這一領域已在多個知名期刊上發表了超過160篇研究文章。他的研究主題包括奇異和退化的非線性偏微分方程、影像處理方程、流量限制擴散方程、質量傳輸理論以及非局部方程。他的一些研究成果已發表在六本專著中：最小線性增長泛函的拋物型準線性方程（Birkhäuser，2023年Ferran Sunyer i Balaguer獎得主）、非局部擴散問題（美國數學學會，2010年）、非局部周長、曲率和最小曲面（Birkhäuser，2019年）、隨機漫步空間中的變分和擴散問題（Birkhäuser，2023年）、最小梯度函數（Birkhäuser，2024年）以及度量測度空間中梯度流的弱解（劍橋大學出版社，2026年）。

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