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This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like applied mathematics, physics, engineering. The main purpose is on the one hand to train the students to appreciate the interplay between theory and modelling in problems arising in the applied sciences; on the other hand to give them a solid theoretical background for numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first one has a rather elementary character with the goal of developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. Ideas and connections with concrete aspects are emphasized whenever possible, in order to provide intuition and feeling for the subject. For this part, a knowledge of advanced calculus and ordinary differential equations is required. Also, the repeated use of the method of separation of variables assumes some basic results from the theory of Fourier series, which are summarized in an appendix. The main topic of the second part is the development of Hilbert space methods for the variational formulation and analysis of linear boundary and initial-boundary value problemsemph{. }% Given the abstract nature of these chapters, an effort has been made to provide intuition and motivation for the various concepts and results. The understanding of these topics requires some basic knowledge of Lebesgue measure and integration, summarized in another appendix. At the end of each chapter, a number of exercises at different level of complexity is included. The most demanding problems are supplied with answers or hints. The exposition if flexible enough to allow substantial changes without compromising the comprehension and to facilitate a selection of topics for a one or two semester course.

Differential equations, Partial. --- Electronic books. -- local. --- Laplacian operator. --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Operator, Laplacian --- Mathematics. --- Partial differential equations. --- Partial Differential Equations. --- Partial differential equations --- Math --- Science --- Differential equations, Partial --- Differential equations, partial.

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Geometry, Differential --- Global analysis (Mathematics) --- Laplacian operator --- Manifolds (Mathematics) --- Géométrie différentielle --- Analyse globale (Mathématiques) --- Variétés (Mathématiques) --- Congresses --- Congrès --- 517.9 --- -Global analysis (Mathematics) --- -Laplacian operator --- -Manifolds (Mathematics) --- -Operator, Laplacian --- Differential equations, Partial --- Differential geometry --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Congresses. --- -Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- -517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Operator, Laplacian --- Géométrie différentielle --- Analyse globale (Mathématiques) --- Variétés (Mathématiques) --- Congrès

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Eigenvectors of graph Laplacians have not, to date, been the subject of expository articles and thus they may seem a surprising topic for a book. The authors propose two motivations for this new LNM volume: (1) There are fascinating subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs. (2) "Geometric" properties of (cost) functions defined on the vertex sets of graphs are of practical interest for heuristic optimization algorithms. The observation that the cost functions of quite a few of the well-studied combinatorial optimization problems are eigenvectors of associated graph Laplacians has prompted the investigation of such eigenvectors. The volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology.

Eigenvectors. --- Laplacian operator. --- Graph theory. --- Vecteurs --- Laplacien --- Théorie des graphes --- Eigenvectors --- Laplacian operator --- Graph theory --- Applied Physics --- Algebra --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Graphs, Theory of --- Theory of graphs --- Operator, Laplacian --- Extremal problems --- Mathematics. --- Algebra. --- Matrix theory. --- Combinatorics. --- Linear and Multilinear Algebras, Matrix Theory. --- Combinatorics --- Mathematical analysis --- Math --- Science --- Combinatorial analysis --- Topology --- Differential equations, Partial --- Matrices --- Vector spaces --- Eigenfactor

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Complex analysis --- Differential geometry. Global analysis --- Operator, Laplacian --- Laplacian operator --- Laplacien --- Riemannian manifolds --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Differential equations, Partial --- Laplacian operator. --- Riemannian manifolds. --- Riemann, Variétés de --- Variétés (mathématiques) --- Équations aux dérivées partielles. --- Differential equations, Partial. --- Laplacien. --- Géometrie différentielle --- Géometrie différentielle --- Variétés (mathématiques)

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The existence, for every sub-Laplacian, of a homogeneous fundamental solution smooth out of the origin, plays a crucial role in the book. This makes it possible to develop an exhaustive Potential Theory, almost completely parallel to that of the classical Laplace operator. This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. In recent years, sub-Laplacian operators have received considerable attention due to their special role in the theory of linear second-order PDE's with semidefinite characteristic form. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra nor in differential geometry. It is thus addressed, besides PhD students, to junior and senior researchers in different areas such as: partial differential equations; geometric control theory; geometric measure theory and minimal surfaces in stratified Lie groups.

Potential theory (Mathematics) --- Harmonic functions. --- Laplacian operator. --- Lie groups. --- Differential equations, Partial. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Partial differential equations --- Operator, Laplacian --- Differential equations, Partial --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Algebra. --- Differential equations, partial. --- Potential theory (Mathematics). --- Topological Groups. --- Partial Differential Equations. --- Potential Theory. --- Topological Groups, Lie Groups. --- Groups, Topological --- Continuous groups --- Mathematics --- Partial differential equations. --- Topological groups.