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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
Dirac’s algorithm in the presence of boundaries: a practical guide to a geometric approach
Bogar Díaz,Juan Margalef-Bentabol,Juan Margalef-Bentabol,Juan Margalef-Bentabol,Eduardo J. S. Villaseñor,Eduardo J. S. Villaseñor +5 more
TL;DR: In this article, the Spanish Ministerio de Ciencia Innovación and Universities-Agencia Estatal de Investigaci´on/FIS2017-84440-C2-2P grant was used to support Bogar Diaz and Juan Margalef-Bentabol.
Journal ArticleDOI
Finite-volume scheme for a degenerate cross-diffusion model motivated from ion transport.
TL;DR: An implicit Euler finite‐volume scheme for a degenerate cross‐diffusion system describing the ion transport through biological membranes and it is shown that the scheme preserves the structure of the equations like nonnegativity, upper bounds, and entropy dissipation.
Journal ArticleDOI
Pullback dynamical behaviors of the non-autonomous micropolar fluid flows
TL;DR: In this article, the pullback asymptotic behaviors of solutions for non-autonomous micropolar fluid flows in two-dimensional bounded domains were investigated, based on the known results concerning the global well-posedness of the solutions.
Journal ArticleDOI
Late-lumping backstepping control of partial differential equations
TL;DR: Three different partial differential equations that can be exponentially stabilized using backstepping controllers are considered and it is proven that the finite-dimensional approximated controller stabilizes the original system if the order is high enough.
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.