Giovanni Leoni - Böcker

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. This book is self-contained. All the statements are fully justified and proved, with the exception of basic results in measure theory, which may be found in any good textbook on the subject. It also contains several exercises. Therefore,it may be used both as a graduate textbook as well as a reference text for researchers in the field. Irene Fonseca is the Mellon College of Science Professor of Mathematics and is currently the Director of the Center for Nonlinear Analysis in the Department of Mathematical Sciences at Carnegie Mellon University.Her research interests lie in the areas of continuum mechanics, calculus of variations, geometric measure theory and partial differential equations. Giovanni Leoni is also a professor in the Department of Mathematical Sciences at Carnegie Mellon University. He focuses his research on calculus of variations, partial differential equations and geometric measure theory with special emphasis on applications to problems in continuum mechanics and in materials science.

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. This book is self-contained. All the statements are fully justified and proved, with the exception of basic results in measure theory, which may be found in any good textbook on the subject. It also contains several exercises. Therefore,it may be used both as a graduate textbook as well as a reference text for researchers in the field. Irene Fonseca is the Mellon College of Science Professor of Mathematics and is currently the Director of the Center for Nonlinear Analysis in the Department of Mathematical Sciences at Carnegie Mellon University.Her research interests lie in the areas of continuum mechanics, calculus of variations, geometric measure theory and partial differential equations. Giovanni Leoni is also a professor in the Department of Mathematical Sciences at Carnegie Mellon University. He focuses his research on calculus of variations, partial differential equations and geometric measure theory with special emphasis on applications to problems in continuum mechanics and in materials science.

This book provides a gentle introduction to fractional Sobolev spaces which play a central role in the calculus of variations, partial differential equations, and harmonic analysis. The first part deals with fractional Sobolev spaces of one variable. It covers the definition, standard properties, extensions, embeddings, Hardy inequalities, and interpolation inequalities.The second part deals with fractional Sobolev spaces of several variables. The author studies completeness, density, homogeneous fractional Sobolev spaces, embeddings, necessary and sufficient conditions for extensions, Gagliardo-Nirenberg type interpolation inequalities, and trace theory. The third part explores some applications: interior regularity for the Poisson problem with the right-hand side in a fractional Sobolev space and some basic properties of the fractional Laplacian.The first part of the book is accessible to advanced undergraduates with a strong background in integration theory; the second part, to graduate students having familiarity with measure and integration and some functional analysis. Basic knowledge of Sobolev spaces would help, but is not necessary. The book can also serve as a reference for mathematicians working in the calculus of variations and partial differential equations as well as for researchers in other disciplines with a solid mathematics background. It contains several exercises and is self-contained.

This book is about differentiation of functions. It is divided into two parts, which can be used as different textbooks, one for an advanced undergraduate course in functions of one variable and one for a graduate course on Sobolev functions. The first part develops the theory of monotone, absolutely continuous, and bounded variation functions of one variable and their relationship with Lebesgue-Stieltjes measures and Sobolev functions. It also studies decreasing rearrangement and curves. The second edition includes a chapter on functions mapping time into Banach spaces. The second part of the book studies functions of several variables. It begins with an overview of classical results such as Rademacher's and Stepanoff's differentiability theorems, Whitney's extension theorem, Brouwer's fixed point theorem, and the divergence theorem for Lipschitz domains. It then moves to distributions, Fourier transforms and tempered distributions. The remaining chapters are a treatise on Sobolev functions. The second edition focuses more on higher order derivatives and it includes the interpolation theorems of Gagliardo and Nirenberg. It studies embedding theorems, extension domains, chain rule, superposition, Poincare's inequalities and traces. A major change compared to the first edition is the chapter on Besov spaces, which are now treated using interpolation theory.

This contributed volume reports on a multidisciplinary collective work on the topic of Smart City, merging scientific reflections and operational issues. Here, current Smart Cities concepts are subjected to criticism, while the related terminology has been updated to contemplate a model of urban development capable of integrating technical and humanistic culture by fostering an open dialogue between different stakeholders. Upon an introduction to the state of the art, this book presents a glossary of definitions and concepts around the contemporary city, and five interviews with researchers and scholars of different background. The last chapter summarizes current challenges in designing the city of the future, highlighting new research directions in home-infrastructure, small smart city, energy transition, connectivity, digitalization and autonomous and connected mobility.Written by the members of the Scientific Committee of the Smart City 4.0 Sustainable LAB ResearchLaboratory, an inter-university network including research groups from the University of Parma, University of Modena and Reggio Emilia, University of Bologna, University of Ferrara, the Polytechnic University of Milan, and the Catholic University of Milan with its Piacenza campus, this book offers a source of inspiration for other researchers and stakeholders, and it is intended to foster collaborations between different stakeholders - and possibly countries – to develop future cities that are wise, green, sustainable and inclusive.

This contributed volume reports on a multidisciplinary collective work on the topic of Smart City, merging scientific reflections and operational issues. Here, current Smart Cities concepts are subjected to criticism, while the related terminology has been updated to contemplate a model of urban development capable of integrating technical and humanistic culture by fostering an open dialogue between different stakeholders. Upon an introduction to the state of the art, this book presents a glossary of definitions and concepts around the contemporary city, and five interviews with researchers and scholars of different background. The last chapter summarizes current challenges in designing the city of the future, highlighting new research directions in home-infrastructure, small smart city, energy transition, connectivity, digitalization and autonomous and connected mobility.Written by the members of the Scientific Committee of the Smart City 4.0 Sustainable LAB ResearchLaboratory, an inter-university network including research groups from the University of Parma, University of Modena and Reggio Emilia, University of Bologna, University of Ferrara, the Polytechnic University of Milan, and the Catholic University of Milan with its Piacenza campus, this book offers a source of inspiration for other researchers and stakeholders, and it is intended to foster collaborations between different stakeholders - and possibly countries – to develop future cities that are wise, green, sustainable and inclusive.