Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure
暫譯: 具有區塊結構的橢圓系統的邊界值問題與哈迪空間

Auscher, Pascal, Egert, Moritz

Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure
  • 出版商: Birkhauser Boston
  • 出版日期: 2024-07-28
  • 售價: $5,540
  • 貴賓價: 9.5 折 $5,263
  • 語言: 英文
  • 頁數: 310
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3031299752
  • ISBN-13: 9783031299759

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商品描述

In this monograph, for elliptic systems with block structure in the upper half-space and t-independent coefficients, the authors settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal ranges of exponents. Prior to this work, only the two-dimensional situation was fully understood. In higher dimensions, partial results for existence in smaller ranges of exponents and for a subclass of such systems had been established. The presented uniqueness results are completely new, and the authors also elucidate optimal ranges for problems with fractional regularity data.

The first part of the monograph, which can be read independently, provides optimal ranges of exponents for functional calculus and adapted Hardy spaces for the associated boundary operator. Methods use and improve, with new results, all the machinery developed over the last two decades to study such problems: the Kato square root estimates and Riesz transforms, Hardy spaces associated to operators, off-diagonal estimates, non-tangential estimates and square functions, and abstract layer potentials to replace fundamental solutions in the absence of local regularity of solutions.

商品描述(中文翻譯)

在這本專著中，針對上半空間中具有區塊結構的橢圓系統及與時間 t 無關的係數，作者解決了邊界值問題的研究，通過證明在最佳指數範圍內的迪里克雷（Dirichlet）、正則性（regularity）和紐曼（Neumann）問題的相容良好定義。在此之前，只有二維情況被完全理解。在更高維度中，已經建立了在較小指數範圍內存在性的部分結果，以及針對這類系統的一個子類的結果。所呈現的唯一性結果是全新的，作者還闡明了具有分數正則性數據的問題的最佳範圍。

專著的第一部分可以獨立閱讀，提供了與相關邊界算子相對應的函數微積分和適應的哈代空間的最佳指數範圍。這些方法使用並改進了過去二十年來為研究這類問題而發展的所有工具，包括卡托平方根估計（Kato square root estimates）和里茲變換（Riesz transforms）、與算子相關的哈代空間、非對角估計（off-diagonal estimates）、非切向估計（non-tangential estimates）和平方函數（square functions），以及抽象層位勢（abstract layer potentials）以替代在缺乏解的局部正則性時的基本解。

作者簡介

​Pascal Auscher is professor of Mathematics in the Laboratoire de Mathématiques d'Orsay at the Université Paris-Saclay. He received his PhD in 1989 at Université Paris-Dauphine under the supervision of Yves Meyer. He is a specialist in harmonic analysis and contributed to the theory of wavelets and to partial differential equations. An outstanding contribution is his participation to the proof of the Kato conjecture in any dimension, which is a starting point for boundary value problems. He has launched a systematic theory of Hardy spaces associated to operators in relation to tent spaces, which is one core of the present monograph. He has recently served as director of the national institute for mathematical sciences and interactions (Insmi) at the national center for scientific research (CNRS).Moritz Egert is professor of Mathematics at the Technical University of Darmstadt. He received his PhD in 2015 in Darmstadt under the supervision ofRobert Haller and was subsequently Maître de Conférences in the Laboratoire de Mathématiques d'Orsay at the Université Paris-Saclay. He is a specialist in harmonic analysis and partial differential equations. In his research, he combines methods from harmonic analysis, operator theory and geometric measure theory to study partial differential equations in non-smooth settings.

作者簡介(中文翻譯)

帕斯卡爾·奧舍爾是巴黎薩克雷大學（Université Paris-Saclay）奧爾塞數學實驗室（Laboratoire de Mathématiques d'Orsay）的數學教授。他於1989年在巴黎多芬大學（Université Paris-Dauphine）獲得博士學位，指導教授為伊夫·梅耶（Yves Meyer）。他專精於調和分析，對小波理論和偏微分方程有重要貢獻。他的一項傑出貢獻是參與了Kato猜想在任意維度的證明，這是邊值問題的起點。他啟動了一個系統的哈代空間（Hardy spaces）理論，該理論與帳篷空間（tent spaces）相關，這是本專著的核心之一。他最近擔任國家科學研究中心（CNRS）數學科學與互動國家研究所（Insmi）的主任。莫里茨·埃格特是達姆施塔特工業大學（Technical University of Darmstadt）的數學教授。他於2015年在達姆施塔特獲得博士學位，指導教授為羅伯特·哈勒（Robert Haller），隨後在巴黎薩克雷大學的奧爾塞數學實驗室擔任講師（Maître de Conférences）。他專精於調和分析和偏微分方程。在他的研究中，他結合了調和分析、算子理論和幾何測度論的方法，以研究非光滑環境中的偏微分方程。

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