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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
Radial solutions of quasilinear equations in Orlicz–Sobolev type spaces
TL;DR: In this paper, the existence of a nonnegative radial solution for the quasilinear elliptic equation is proved based on variational methods in the Orlicz-Sobolev spaces.
Journal ArticleDOI
Projection-Based Model Reduction with Dynamically Transformed Modes
TL;DR: In this paper, a new model reduction framework for problems that exhibit transport phenomena is proposed, which employs time-dependent transformation operators and generalizes MFEM to arbitrary basis functions, which is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions.
Posted Content
Solvability of a Keller-Segel system with signal-dependent sensitivity and essentially sublinear production
TL;DR: In this paper, the authors considered the zero-flux chemotaxis system and proved that no chemotactic collapse for the cell distribution occurs in the sense that any arbitrary nonnegative and sufficiently regular initial data $u(x,0) emanates a unique pair of global and uniformly bounded functions $(u,v) which classically solve the corresponding initial-boundary value problem.
Journal ArticleDOI
Poiseuille Flow of Nematic Liquid Crystals via the Full Ericksen–Leslie Model
TL;DR: In this article, the authors studied the Cauchy problem of the Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals and established the global existence of weak solutions that are continuous and have bounded energy.
Proceedings ArticleDOI
Existence of solutions to chemotaxis dynamics with logistic source
Tomomi Yokota,Noriaki Yoshino +1 more
TL;DR: In this paper, a chemotaxis system with nonlinear diffusion and logistic growth term was considered and it was shown that there exists a local solution to this system for any $L 2$-initial data.
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.