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Functional Analysis, Sobolev Spaces and Partial Differential Equations
TLDR
In this article, the theory of conjugate convex functions is introduced, and the Hahn-Banach Theorem and the closed graph theorem are discussed, as well as the variations of boundary value problems in one dimension.Abstract:
Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.read more
Citations
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Journal ArticleDOI
Robust and optimal multi-iterative techniques for IgA Galerkin linear systems
Marco Donatelli,Carlo Garoni,Carla Manni,Stefano Serra-Capizzano,Hendrik Speleers,Hendrik Speleers +5 more
TL;DR: This work considers fast solvers for the large linear systems coming from the Isogeometric Analysis (IgA) collocation approximation based on B-splines of full elliptic d-dimensional Partial Differential Equations (PDEs) and designs iterative algorithms which are optimal and robust.
Journal ArticleDOI
On a universal solution to the transport-of-intensity equation.
TL;DR: In this article, a universal solution to TIE with the advantages of high accuracy, convergence guarantee, applicability to arbitrarily shaped regions, and simplified implementation and computation is proposed, where the maximum intensity assumption is removed and the initial solution is further refined iteratively by solving the same Poisson equation.
Journal ArticleDOI
Constructive Approximation by Superposition of Sigmoidal Functions
Danilo Costarelli,Renato Spigler +1 more
TL;DR: In this paper, a constructive theory for approximating func- tions of one or more variables by superposition of sigmoidal functions is developed, which is done in the uniform norm as well as in the L p norm.
Journal ArticleDOI
Existence of solutions to boundary value problem for impulsive fractional differential equations
TL;DR: In this paper, the existence and multiplicity of solutions for an impulsive boundary value problem for fractional order differential equations are studied and the notions of classical and weak solutions are introduced.
Journal ArticleDOI
Persistence criteria for populations with non-local dispersion
TL;DR: It is proved that a positive stationary solution exists if and only if the generalised principal eigenvalue of the linear problemλp is a spectral quantity that is defined in the spirit of the generalisation first eigen value of an elliptic operator.
References
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Book
Linear and Quasilinear Equations of Parabolic Type
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
Book
Non-homogeneous boundary value problems and applications
TL;DR: In this paper, the authors consider the problem of finding solutions to elliptic boundary value problems in Spaces of Analytic Functions and of Class Mk Generalizations in the case of distributions and Ultra-Distributions.
Book
Introduction to Fourier Analysis on Euclidean Spaces.
Elias M. Stein,Guido Weiss +1 more
TL;DR: In this paper, the authors present a unified treatment of basic topics that arise in Fourier analysis, and illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations.
Book
Theory of function spaces
TL;DR: In this article, the authors measure smoothness using Atoms and Pointwise Multipliers, Wavelets, Spaces on Lipschitz Domains, Wavelet and Sampling Numbers.