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This monograph presents an introduction to some geometric and analytic aspects of the maximum principle. In doing so, it analyses with great detail the mathematical tools and geometric foundations needed to develop the various new forms that are presented in the first chapters of the book. In particular, a generalization of the Omori-Yau maximum principle to a wide class of differential operators is given, as well as a corresponding weak maximum principle and its equivalent open form and parabolicity as a special stronger formulation of the latter.  In the second part, the attention focuses on a wide range of applications, mainly to geometric problems, but also on some analytic (especially PDEs) questions including: the geometry of submanifolds, hypersurfaces in Riemannian and Lorentzian targets, Ricci solitons, Liouville theorems, uniqueness of solutions of Lichnerowicz-type PDEs and so on. Maximum Principles and Geometric Applications is written in an easy style making it accessible to beginners. The reader is guided with a detailed presentation of some topics of Riemannian geometry that are usually not covered in textbooks. Furthermore, many of the results and even proofs of known results are new and lead to the frontiers of a contemporary and active field of research.

Geometry --- Mathematics --- Physical Sciences & Mathematics --- Maximum principles (Mathematics) --- Geometric analysis. --- Geometric analysis PDEs (Geometric partial differential equations) --- Mathematical analysis --- Differential equations, Partial --- Numerical solutions --- Global analysis. --- Differential equations, partial. --- Geometry. --- Global Analysis and Analysis on Manifolds. --- Partial Differential Equations. --- Euclid's Elements --- Partial differential equations --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Partial differential equations. --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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This book collects recent research papers by respected specialists in the field. It presents advances in the field of geometric properties for parabolic and elliptic partial differential equations, an area that has always attracted great attention. It settles the basic issues (existence, uniqueness, stability and regularity of solutions of initial/boundary value problems) before focusing on the topological and/or geometric aspects. These topics interact with many other areas of research and rely on a wide range of mathematical tools and techniques, both analytic and geometric. The Italian and Japanese mathematical schools have a long history of research on PDEs and have numerous active groups collaborating in the study of the geometric properties of their solutions. .

Mathematics. --- Functional analysis. --- Differential equations. --- Partial differential equations. --- Convex geometry. --- Discrete geometry. --- Calculus of variations. --- Partial Differential Equations. --- Functional Analysis. --- Ordinary Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Convex and Discrete Geometry. --- Differential equations, Parabolic --- Geometric analysis --- Differential equations, Elliptic --- Geometric analysis PDEs (Geometric partial differential equations) --- Parabolic differential equations --- Parabolic partial differential equations --- Geometry --- Mathematical analysis --- Differential equations, Partial --- Differential equations, partial. --- Differential Equations. --- Mathematical optimization. --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- 517.91 Differential equations --- Differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations --- Discrete mathematics --- Convex geometry . --- Isoperimetrical problems --- Variations, Calculus of --- Combinatorial geometry