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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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Existence of positive solutions for a class of quasilinear elliptic problems with exponential growth via the Nehari manifold method
TL;DR: In this article, a superlinear continuous function with exponential subcritical or exponential critical growth is considered and the problem is solved using the Nehari manifold method, and the results include a large class of problems.
Journal ArticleDOI
Existence of solutions of the abstract Cauchy problem of fractional order
TL;DR: In this article, the authors studied the differentiability of mild solutions for a class of fractional abstract Cauchy problems and established the existence of classical solutions for the homogeneous Cauche problem in terms of the α-resolvent family corresponding to the problem.
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Identification of time-dependent convection coefficient in a time-fractional diffusion equation
TL;DR: The existence, uniqueness, regularity, and uniqueness of solution for the direct problem are proved by using the fixed point theorem and the stability of inverse convection coefficient problem is obtained based on the regularity of Solution for thedirect problem and some generalized Gronwall’s inequalities.
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Sobolev spaces and calculus of variations on fractals
TL;DR: In this paper, a review of $p$-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form is presented.
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Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity
TL;DR: In this article, the authors investigate the parabolic-elliptic Keller-Segel model and show that whenever the initial data satisfy only certain requirements on regularity and on positivity, one can find at least one global generalized solution.
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.
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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.
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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.
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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.