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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
A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data.
TL;DR: A variational model for learning convolutional image atoms from corrupted and/or incomplete data is introduced and analyzed both in function space and numerically, and fundamental analytical properties allowing well-posedness and stability results for inverse problems are proven in a continuous setting.
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Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods
Christian Kanzow,Daniel Steck +1 more
TL;DR: An algorithm of augmented Lagrangian type is constructed which reduces the QVI to a sequence of standard variational inequalities and a full convergence analysis is provided which includes the existence of solutions of the subproblems as well as the attainment of feasibility and optimality.
Journal ArticleDOI
Existence and multiplicity of solutions for a class of ( ź 1 , ź 2 )-Laplacian elliptic system in R N via genus theory
TL;DR: By using the least action principle, it is obtained that a nonlinear and non-homogeneous elliptic system involving ( ź 1, ź 2 ) -Laplacian has at least one nontrivial solution.
Existence and decay of solutions to a viscoelastic plate equation
TL;DR: In this article, the authors studied the fourth-order viscoelastic plate equation with non-traditional boundary conditions and established the well-posedness and a decay result for the decay.
Journal ArticleDOI
Entropy formulation of degenerate parabolic equation with zero-flux boundary condition
TL;DR: In this paper, the authors considered the degenerate hyperbolic-parabolic equation and proved existence of entropy solution for any space dimension N ≥ 1 under a partial genuine nonlinearity assumption on f.
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.