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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
Multi-bang control of elliptic systems
Christian Clason,Karl Kunisch +1 more
TL;DR: In this article, a primal-dual optimality system with Fenchel duality was proposed for multi-bang control problems, where a distributed control should only take on values from a discrete set of allowed states.
Journal ArticleDOI
Distributed stabilization of Korteweg–de Vries–Burgers equation in the presence of input delay
Wen Kang,Wen Kang,Emilia Fridman +2 more
TL;DR: A new Lyapunov–Krasovskii functional is suggested that leads to regional stability conditions of the closed-loop system in terms of linear matrix inequalities (LMIs) by solving these LMIs and an upper bound on the delay that preserves regional stability can be found.
Posted Content
Representation formulas and pointwise properties for Barron functions
Weinan E,Stephan Wojtowytsch +1 more
TL;DR: It is shown that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm.
Journal ArticleDOI
On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs
TL;DR: In this article, the authors consider two classes of PDEs, linear second-order elliptic and parabolic, and show that the sequence of minimizers strongly converges to the PDE solution in $C^0$.
Journal ArticleDOI
Data-adaptive harmonic spectra and multilayer Stuart-Landau models.
TL;DR: In this article, a data-adaptive harmonic decomposition of multivariate time series is considered for which an integral operator approach with periodic semigroup kernels is adopted for multilayer Stuart-Landau models.
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.