This book investigates the close relation between quite sophisticated function spaces, the regularity of solutions of partial differential equations (PDEs) in these spaces and the link with the numerical solution of such PDEs. It consists of three parts. Part I, the introduction, provides a quick guide to function spaces and the general concepts needed. Part II is the heart of the monograph and deals with the regularity of solutions in Besov and fractional Sobolev spaces. In particular, it studies regularity estimates of PDEs of elliptic, parabolic and hyperbolic type on non smooth domains. Linear as well as nonlinear equations are considered and special attention is paid to PDEs of parabolic type. For the classes of PDEs investigated a justification is given for the use of adaptive numerical schemes. Finally, the last part has a slightly different focus and is concerned with traces in several function spaces such as Besov– and Triebel–Lizorkin spaces, but also in quite general smoothness Morrey spaces.

The book is aimed at researchers and graduate students working in regularity theory of PDEs and function spaces, who are looking for a comprehensive treatment of the above listed topics.

Table of Contents

1 Introduction

2 Function Spaces and General Concepts

Part I Besov and Fractional Sobolev Regularity of PDEs

3 Theory and Background Material for PDEs

4 Regularity Theory for Elliptic PDEs

5 Regularity Theory for Parabolic PDEs

6 Regularity Theory for Hyperbolic PDEs

7 Applications to Adaptive Approximation Schemes

Part II Traces in Function Spaces

8 Traces on Lipschitz Domains

9 Traces of Generalized Smoothness Morrey Spaces on Domains

10 Traces on Riemannian Manifolds

About the Author

Cornelia Schneider is Senior Lecturer at the Friedrich-Alexander University of Erlangen-Nuremberg, Germany, where she has taught since 2010. Her research interests are in Applied Analysis, in particular, regularity theory of PDEs and function spaces. From 2000-2006 she studied Mathematics with minor Physics at the Friedrich-Schiller University in Jena (spending 1 year abroad in Australia and New Zealand) and obtained her PhD at the University of Leipzig in 2009. From 2009-2010 she worked as a postdoc at the University of Coimbra in Portugal. In 2020 she was awarded the Habilitation as the first woman in Mathematics at the Philipps-University of Marburg.