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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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Sensitivity equation method for the Navier-Stokes equations applied to uncertainty propagation
TL;DR: In this article, a finite element-volume numerical scheme for the state and the sensitivity of the Navier-Stokes equations is proposed, which is integrated into the open-source industrial code TrioCFD.
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Phase separation patterns from directional quenching
Rafael Monteiro,Arnd Scheel +1 more
TL;DR: This work model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across a trigger line, and finds existence and nonexistence results of single interfaces and striped patterns.
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Hölder continuity of the solution map to an elliptic optimal control problem with mixed constraints
Vu Huu Nhu,N. H. Auh,B. T. Kien +2 more
TL;DR: In this paper, the authors investigate the Holder continuity of the solution map to a parametric optimal control problem, which is governed by elliptic equations with mixed control-state constraints and convex cost functions.
Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences
TL;DR: The central limit theorem of the Sinkhorn potentials and the weak limits of the couplings are obtained, proving a conjecture of Harchaoui, Liu and Pal (2020) and enabling statistical inference based on entropic regularized optimal transport.
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Analysis and boundary value problems on singular domains: An approach via bounded geometry
TL;DR: In this paper, the authors prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains that are conformally equivalent to manifolds with boundary and bounded geometry.
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.