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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
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Stochastic porous media equations in Rd
TL;DR: In this paper, the existence and uniqueness of solutions to the stochastic porous media equation d X − Δ ψ ( X ) d t = X d W in R d are studied.
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Weak Compactness Techniques and Coagulation Equations
TL;DR: In particular, the use of weak L 1-compactness techniques led to a mature theory of weak solutions and the purpose of these notes is to describe the results obtained so far in that direction, as well as the mathematical tools used as mentioned in this paper.
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Applications of local linking to nonlocal Neumann problems
TL;DR: In this article, a non-local Neumann problem driven by a nonhomogeneous elliptic differential operator is studied, where the reaction term is a nonlinearity function that exhibits psuperlinear growth but need not satisfy the Ambrosetti-Rabinowitz condition.
Journal ArticleDOI
Analysis of a conforming finite element method for the Boussinesq problem with temperature-dependent parameters
TL;DR: This paper analyzes a conforming finite element method for the numerical simulation of non-isothermal incompressible fluid flows subject to a heat source modeled by a generalized Boussinesq problem with temperature-dependent parameters, and derives optimal a priori error estimates.
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Convergence to equilibrium for positive solutions of some mutation-selection model
TL;DR: In this article, the authors studied the long time behavior of the positive solutions of the mutation selection model with Neumann boundary condition and showed that for sufficiently small constant values of the Neumann bound, there exists a unique positive steady state which is positively globally stable.
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.