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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
Approximate controllability for nonlinear degenerate parabolic problems with bilinear control
TL;DR: It is shown that the above system can be steered in L^2(\Omega) from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear static controls.
Journal ArticleDOI
Coupling of the Crank–Nicolson scheme and localized meshless technique for viscoelastic wave model in fluid flow
O. Nikan,Zakieh Avazzadeh +1 more
TL;DR: An efficient localized meshless technique for approximating the viscoelastic wave model by decomposing the initial domain into several sub-domains and constructing a local radial basis function approximation over every sub-domain is proposed.
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Approximate controllability for linear degenerate parabolic problems with bilinear control
TL;DR: In this paper, the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy-Neumann problems was studied, and it was shown that the above systems can be steered in L 2 from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target state by bilinear piecewise static controls.
Maximal regularity for evolution equations governed by non-autonomous forms
TL;DR: In this article, a non-autonomous evolutionary problem with maximal regularity was studied and well-posedness was shown for piecewise Lipschitz-continuous and symmetric forms.
Journal ArticleDOI
On a model of a population with variable motility
TL;DR: In this paper, a reaction-diffusion equation with a nonlocal reaction term was studied and a global supremum bound for solutions of the equation was established, and the asymptotic behavior of the population with variable motility was investigated.
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.