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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
Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs
Mickaël D. Chekroun,Honghu Liu +1 more
TL;DR: In this article, the authors proposed a new approach for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type, where a PM provides an approximate parameterization of the high modes by the controlled low ones so that the unexplained high-mode energy is reduced.
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Carleman estimates and null controllability for a degenerate population model
TL;DR: In this paper, a degenerate model describing the dynamics of a population depending on time, on age and on space is presented. But the degeneracy can occur at the boundary or in the interior of the space domain and the authors focus on null controllability problem.
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Bifurcation properties for a class of fractional Laplacian equations in
TL;DR: In this paper, the Leray-Shauder degree theory and the global bifurcation result due to Rabinowitz were used to study the bifurbation properties of nonlocal problems.
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Fully spectral collocation method for nonlinear parabolic partial integro-differential equations
TL;DR: In this paper, the spectral approximation of nonlinear parabolic Volterra and Fredholm partial integro-differential equations is studied using the spectral method for the purpose of discretizing the time variable by finite difference schemes.
Journal ArticleDOI
Poincaré inequalities for Sobolev spaces with matrix-valued weights and applications to degenerate partial differential equations
TL;DR: For bounded domains Ω, this article showed that the Lp-norm of a regular function with compact support is controlled by weighted Lpnorms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions.
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.