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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
Conjugate Gradient Method in Hilbert and Banach Spaces to Enhance the Spatial Resolution of Radiometer Data
TL;DR: A new computer-time effective iterative method is proposed to enhance the spatial resolution of microwave radiometer data in both Hilbert and Banach spaces and is able to provide a reconstruction accuracy similar to the conventional Landweber method, but with a significantly reduced processing time.
Journal ArticleDOI
Simultaneous deconvolution and denoising using a second order variational approach applied to image super resolution
TL;DR: This work proposes a novel multiframe image SR algorithm based on a convex combination of Bilateral Total Variation and a non-smooth second order variational regularization, using a controlled weighting parameter and proves the existence of a minimizer of the proposed energy in the space of functions of bounded Hessian.
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Beurling–Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations
TL;DR: In this paper, the authors obtained a boundedness characterization of the Beurling-Ahlfors commutator in the weighted Morrey space and applied it to the Beltrami equation.
Journal ArticleDOI
Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
TL;DR: In this article, a Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space was shown to generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Simon (1983).
Posted Content
Stable solutions to semilinear elliptic equations are smooth up to dimension 9
TL;DR: In this paper, it was shown that stable solutions to semilinear elliptic equations are smooth in dimension n ≥ 9, where n is the number of vertices in the graph.
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.