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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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Approximation of the Fokker-Planck equation of the stochastic chemostat
TL;DR: An adapted finite difference scheme is proposed for the approximation of the solution of the Fokker-Planck equation with respect to the washout of the two-dimensional chemostat.
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Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model
TL;DR: In this article, the existence of weak solutions for a thermodynamically consistent phase-field model in two and three dimensions of space is proved. But the authors do not consider the case where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality.
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Lower semicontinuity of the solution map to a parametric elliptic optimal control problem with mixed pointwise constraints
B. T. Kien,Vu Huu Nhu,Arnd Rösch +2 more
TL;DR: In this article, the authors study the stability of a parametric optimal control problem with mixed control-state constraints and convex cost functions, and obtain sufficient conditions under which the solution map to an elliptic optimal control.
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Evolution Equations governed by Lipschitz Continuous Non-autonomous Forms
Ahmed Sani,Hafida Laasri +1 more
TL;DR: In this paper, the authors proved the maximal regularity of linear non-autonomous evolutionary Cauchy problems with time Lipschitz continuous operator. But they used the frozen coefficient method to obtain an invariance criterion for convex and closed sets.
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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.