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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
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
Sachiko Ishida,Tomomi Yokota +1 more
TL;DR: In this article, the authors gave a blow-up result for the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with large negative energy.
Journal ArticleDOI
Quantization and clustering with Bregman divergences
TL;DR: A quantization scheme with a class of distortion measures called Bregman divergences is used, and conditions ensuring the existence of an optimal quantizer and an empirically optimal quantizers are provided.
Journal ArticleDOI
Some A priori estimates for the homogeneous Landau equation with soft potentials
TL;DR: In this article, a priori estimates for the homogeneous Landau equation with soft potentials were obtained for the Coulomb case, where the control on time integral of some weighted Fisher information is required.
Journal ArticleDOI
Persistence versus extinction under a climate change in mixed environments
TL;DR: In this article, the authors studied the persistence versus extinction of species in the reaction diffusion equation, where the environment is only assumed to be globally unfavorable with favorable pockets extending to infinity, and the reaction term is time-independent or time-periodic dependent.
Posted Content
Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems
Christoph Reisinger,Yufei Zhang +1 more
TL;DR: In this article, it was shown that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy.
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.