The use of typical tools from Analysis to shed light on geometric properties is at the core of modern Global Differential Geometry. The aim of the book is to discuss the influence of the structure of a manifold M on the qualitative behavior of solutions to certain classes of elliptic PDE’s arising in a Riemannian context. The differential inequalities that we consider are elliptic, possibly degenerate, with a left-hand side described by a quasilinear operator, which encompasses well known examples, including for instance the p-Laplacian and the mean curvature operator. To illustrate our results, in the first two chapters we have chosen to discuss some geometric and physical problems where such inequalities naturally appear, and in what follows we shall often address special attention to the mean curvature operator, that arises when considering the graph associated to a smooth function u on the manifold. This prototype case leads to investigate a large class of differential inequalities whose right-hand side not only depends on x of M and on the solution u, but also on the gradient of u. Having motivated our study, a key part of the book deals with various generalized forms of the maximum principle, that serve as a bridge to relate the Geometry of M to the analytical properties of the PDE under consideration. Our technical achievements provide a bulk of flexible methods to describe qualitative properties of solutions to a variety of problems. In the last chapters, Liouville type theorems and compact support principles are investigated in detail, with special emphasis on the role played by the integrability requirements that, in the literature, are known as the Keller-Osserman conditions. Geometric applications, among others, touch upon Bernstein type theorems for graphs with prescribed mean curvature (notably, minimal or solitons for the mean curvature flow), Yamabe and capillarity equations. The book presents the most recent points of view and trends in the field, and is therefore to be considered at an advanced level suitable to researchers and senior post-graduate students with a solid knowledge in Differential Geometry and elliptic PDE’s. Besides the presentation of new theorems, we collect and organize sharp refinements of some research results obtained by the authors and their collaborators in a period of about twenty years. The monograph fulfills two purposes. First, it provides an exposition of some fundamental tools in Geometric Analysis and elliptic PDE’s on complete manifolds, a subject that has become of general interest in the last decades, second it also enables researchers to get familiar with the necessary results and concepts to proceed towards a further study of the specialized literature on the subject. We include an extensive and commented bibliography in the attempt to make it a reference book for future research.
Geometric Analysis of Quasilinear Inequalities on Complete Manifolds
Pucci, Patrizia
;
2021
Abstract
The use of typical tools from Analysis to shed light on geometric properties is at the core of modern Global Differential Geometry. The aim of the book is to discuss the influence of the structure of a manifold M on the qualitative behavior of solutions to certain classes of elliptic PDE’s arising in a Riemannian context. The differential inequalities that we consider are elliptic, possibly degenerate, with a left-hand side described by a quasilinear operator, which encompasses well known examples, including for instance the p-Laplacian and the mean curvature operator. To illustrate our results, in the first two chapters we have chosen to discuss some geometric and physical problems where such inequalities naturally appear, and in what follows we shall often address special attention to the mean curvature operator, that arises when considering the graph associated to a smooth function u on the manifold. This prototype case leads to investigate a large class of differential inequalities whose right-hand side not only depends on x of M and on the solution u, but also on the gradient of u. Having motivated our study, a key part of the book deals with various generalized forms of the maximum principle, that serve as a bridge to relate the Geometry of M to the analytical properties of the PDE under consideration. Our technical achievements provide a bulk of flexible methods to describe qualitative properties of solutions to a variety of problems. In the last chapters, Liouville type theorems and compact support principles are investigated in detail, with special emphasis on the role played by the integrability requirements that, in the literature, are known as the Keller-Osserman conditions. Geometric applications, among others, touch upon Bernstein type theorems for graphs with prescribed mean curvature (notably, minimal or solitons for the mean curvature flow), Yamabe and capillarity equations. The book presents the most recent points of view and trends in the field, and is therefore to be considered at an advanced level suitable to researchers and senior post-graduate students with a solid knowledge in Differential Geometry and elliptic PDE’s. Besides the presentation of new theorems, we collect and organize sharp refinements of some research results obtained by the authors and their collaborators in a period of about twenty years. The monograph fulfills two purposes. First, it provides an exposition of some fundamental tools in Geometric Analysis and elliptic PDE’s on complete manifolds, a subject that has become of general interest in the last decades, second it also enables researchers to get familiar with the necessary results and concepts to proceed towards a further study of the specialized literature on the subject. We include an extensive and commented bibliography in the attempt to make it a reference book for future research.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
