The main goal of the book is to provide a comprehensive and self-contained proof of the, relatively recent, theorem of characterization of the strong maximum principle due to Molina-Meyer and the author, published in Diff. Int. Eqns. in 1994, which was later refined by Amann and the author in a paper published in J. of Diff. Eqns. in 1998. Besides this characterization has been shown to be a pivotal result for the development of the modern theory of spatially heterogeneous nonlinear elliptic and parabolic problems; it has allowed us to update the classical theory on the maximum and minimum principles by providing with some extremely sharp refinements of the classical results of Hopf and Protter-Weinberger. By a celebrated result of Berestycki, Nirenberg and Varadhan, Comm. Pure Appl. Maths. in 1994, the characterization theorem is partially true under no regularity constraints on the support domain for Dirichlet boundary conditions.Instead of encyclopedic generality, this book pays special attention to completeness, clarity and transparency of its exposition so that it can be taught even at an advanced undergraduate level. Adopting this perspective, it is a textbook; however, it is simultaneously a research monograph about the maximum principle, as it brings together for the first time in the form of a book, the most paradigmatic classical results together with a series of recent fundamental results scattered in a number of independent papers by the author of this book and his collaborators.Chapters 3, 4, and 5 can be delivered as a classical undergraduate, or graduate, course in Hilbert space techniques for linear second order elliptic operators, and Chaps. 1 and 2 complete the classical results on the minimum principle covered by the paradigmatic textbook of Protter and Weinberger by incorporating some recent classification theorems of supersolutions by Walter, 1989, and the author, 2003. Consequently, these five chapters can be taught at an undergraduate, or graduate, level. Chapters 6 and 7 study the celebrated theorem of Krein-Rutman and infer from it the characterizations of the strong maximum principle of Molina-Meyer and Amann, in collaboration with the author, which have been incorporated to a textbook by the first time here, as well as the results of Chaps. 8 and 9, polishing some recent joint work of Cano-Casanova with the author. Consequently, the second half of the book consists of a more specialized monograph on the maximum principle and the underlying principal eigenvalues.
This Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure set of zeroes of a general class of nonlinear operators. Appealing to a broad audience, it contains many important contributions to linear algebra, linear functional analysis, nonlinear functional analysis, and topology. The author gives several applications of the abstract theory to reaction diffusion equations and systems. The results presented cover a thirty-year period and cut across a variety of mathematical fields.
What should happen when doctors and parents disagree about what would be best for a child? When should courts become involved? Should life support be stopped against parents’ wishes? The case of Charlie Gard, reached global attention in 2017. It led to widespread debate about the ethics of disagreements between doctors and parents, about the place of the law in such disputes, and about the variation in approach between different parts of the world. In this book, medical ethicists Dominic Wilkinson and Julian Savulescu critically examine the ethical questions at the heart of disputes about medical treatment for children. They use the Gard case as a springboard to a wider discussion about the rights of parents, the harms of treatment, and the vital issue of limited resources. They discuss other prominent UK and international cases of disagreement and conflict. From opposite sides of the debate Wilkinson and Savulescu provocatively outline the strongest arguments in favour of and against treatment. They analyse some of the distinctive and challenging features of treatment disputes in the 21st century and argue that disagreement about controversial ethical questions is both inevitable and desirable. They outline a series of lessons from the Gard case and propose a radical new ‘dissensus’ framework for future cases of disagreement. This new book critically examines the core ethical questions at the heart of disputes about medical treatment for children. The contents review prominent cases of disagreement from the UK and internationally and analyse some of the distinctive and challenging features around treatment disputes in the 21st century. The book proposes a radical new framework for future cases of disagreement around the care of gravely ill people.
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