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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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Feedback boundary stabilization to trajectories for 3D Navier-Stokes equations
TL;DR: In this paper, a finite-dimensional feedback boundary control that stabilizes the linear Oseen-Stokes system around the given trajectory to zero is presented. But it is not shown that the same controller also stabilizes, locally, the Navier Stokes system to a given trajectory.
Journal ArticleDOI
Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions
Juntang Ding,Wei Kou +1 more
TL;DR: In this paper, a nonlinear reaction diffusion equation with nonlocal boundary conditions is investigated and the authors derive that the solution blows up at some finite time, and upper and lower bounds of the blow-up time are obtained.
Journal ArticleDOI
Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls
Tatsien Li,Bopeng Rao +1 more
TL;DR: In this article, the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls is shown. But the authors do not consider the non-exact control of the initial data with the same level of energy.
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Morse theory and local linking for a nonlinear degenerate problem arising in the theory of electrorheological fluids
TL;DR: In this article, the nonlinear stationary equation is studied under Dirichlet boundary conditions, where Ω is a smooth bounded domain in R n, p > 1 is a continuous function, and f ( x, u ) has a sublinear growth near the origin.
Journal ArticleDOI
Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations.
TL;DR: This work proves under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization.
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.