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Ebook Partial differential equations in action: Part 2

(BQ) Part 2 book "Partial differential equations in action" has contents: Distributions and sobolev spaces, elements of functional analysis, variational formulation of elliptic problems, elements of functional analysis. | 6 Elements of Functional Analysis Motivations – Norms and Banach Spaces – Hilbert Spaces – Projections and Bases – Linear Operators and Duality – Abstract Variational Problems – Compactness and Weak Convergence – The Fredholm Alternative – Spectral Theory for Symmetric Bilinear Forms 6.1 Motivations The main purpose in the previous chapters has been to introduce part of the basic and classical theory of some important equations of mathematical physics. The emphasis on phenomenological aspects and the connection with a probabilistic point of view should have conveyed to the reader some intuition and feeling about the interpretation and the limits of those models. The few rigorous theorems and proofs we have presented had the role of bringing to light the main results on the qualitative properties of the solutions and justifying, partially at least, the well-posedness of the relevant boundary and initial/boundary value problems we have considered. However, these purposes are somehow in competition with one of the most important role of modern mathematics, which is to reach a unifying vision of large classes of problems under a common structure, capable not only of increasing theoretical understanding, but also of providing the necessary flexibility to guide the numerical methods which will be used to compute approximate solutions. This conceptual jump requires a change of perspective, based on the introduction of abstract methods, historically originating from the vain attempts to solve basic problems (e.g. in electrostatics) at the end of the 19th century. It turns out that the new level of knowledge opens the door to the solution of complex problems in modern technology. These abstract methods, in which analytical and geometrical aspects fuse, are the core of the branch of Mathematics, called Functional Analysis. Salsa S. Partial Differential Equations in Action: From Modelling to Theory c Springer-Verlag 2008, Milan 6.1 Motivations 303 It could be useful for .