A Primer on Semiconvex Functions in General Potential Theories
暫譯: 一般潛能理論中的半凸函數入門指南

Payne, Kevin R., Redaelli, Davide Francesco

A Primer on Semiconvex Functions in General Potential Theories
  • 出版商: Springer
  • 出版日期: 2025-08-07
  • 售價: $3,380
  • 貴賓價: 9.5 折 $3,211
  • 語言: 英文
  • 頁數: 141
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3031943392
  • ISBN-13: 9783031943393
  • 相關分類: 離散數學 Discrete-mathematics

    • 海外代購書籍(需單獨結帳)

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商品描述

This book examines the symbiotic interplay between fully nonlinear elliptic partial differential equations and general potential theories of second order. Starting with a self-contained presentation of the classical theory of first and second order differentiability properties of convex functions, it collects a wealth of results on how to treat second order differentiability in a pointwise manner for merely semicontinuous functions. The exposition features an analysis of upper contact jets for semiconvex functions, a proof of the equivalence of two crucial, independently developed lemmas of Jensen (on the viscosity theory of PDEs) and Slodkowski (on pluripotential theory), and a detailed description of the semiconvex approximation of upper semicontinuous functions.

The foundations of general potential theories are covered, with a review of monotonicity and duality, and the basic tools in the viscosity theory of generalized subharmonics, culminating in an account of the monotonicity-duality method for proving comparison principles. The final section shows that the notion of semiconvexity extends naturally to manifolds. A complete treatment of important background results, such as Alexandrov's theorem and a Lipschitz version of Sard's lemma, is provided in two appendices.

The book is aimed at a wide audience, including professional mathematicians working in fully nonlinear PDEs, as well as master's and doctoral students with an interest in mathematical analysis.

商品描述(中文翻譯)

本書探討完全非線性橢圓偏微分方程與二階一般勢理論之間的共生互動。首先以自成一體的方式介紹凸函數的一階和二階可微性理論，並收集大量結果，說明如何以點wise的方式處理僅半連續函數的二階可微性。內容包括對半凸函數的上接觸噴流的分析，證明Jensen（關於偏微分方程的粘性理論）和Slodkowski（關於多重勢理論）兩個關鍵獨立發展的引理的等價性，以及對上半連續函數的半凸近似的詳細描述。

本書涵蓋一般勢理論的基礎，回顧單調性和對偶性，以及廣義次調和函數的粘性理論中的基本工具，最終介紹了用於證明比較原則的單調性-對偶性方法。最後一部分顯示半凸性概念自然地擴展到流形上。附錄中提供了重要背景結果的完整處理，例如Alexandrov定理和Sard引理的Lipschitz版本。

本書的讀者對象廣泛，包括從事完全非線性偏微分方程的專業數學家，以及對數學分析感興趣的碩士和博士生。

作者簡介

Kevin R. Payne is a full professor of mathematical analysis at the Università di Milano and a Fellow of the American Mathematical Society. He received a BA in Mathematics at Rice University and a PhD in Mathematics at Stony Brook University. He is the author of the book Comparison Principles for General Potential Theories and PDEs (Annals of Mathematics Studies 218, Princeton University Press) with Marco Cirant, Reese Harvey and Blaine Lawson.

Davide Francesco Redaelli is a post-doc research associate in mathematics at the University of Rome Tor Vergata. He received a MSc in Mathematics at the University of Milan, under the supervision of Prof. Kevin R. Payne, and a PhD in Mathematics at the University of Padua, under the supervision of Prof. Marco Cirant. These Lecture Notes are an improved version of the main chapters of his Master's Thesis.

作者簡介(中文翻譯)

Kevin R. Payne 是米蘭大學的數學分析全職教授，也是美國數學學會的會士。他在萊斯大學獲得數學學士學位，並在石溪大學獲得數學博士學位。他與 Marco Cirant、Reese Harvey 和 Blaine Lawson 共同撰寫了《一般潛能理論與偏微分方程的比較原則》（《數學研究年鑑》218，普林斯頓大學出版社）。

Davide Francesco Redaelli 是羅馬托爾維爾大學的數學博士後研究助理。他在米蘭大學獲得數學碩士學位，指導教授為 Kevin R. Payne，並在帕多瓦大學獲得數學博士學位，指導教授為 Marco Cirant。這些講義是他碩士論文主要章節的改進版本。

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