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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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EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity
Hamed Azizollahi,Marion Darbas,Mohamadou Malal Diallo,Abdellatif El Badia,Stephanie Lohrengel +4 more
TL;DR: This study attests that the presence of fontanels in neonates does have an impact on EEG measurements, and a model for the forward problem in EEG source localization is proposed, able to take into account the presence and ossification process offontanels which are characterized by a variable conductivity.
Journal ArticleDOI
Coupled Modeling of Multiphase Flow and Fault Poromechanics During Geologic CO2 Storage
Birendra Jha,Ruben Juanes +1 more
TL;DR: In this paper, the authors presented a new computational approach to model coupled multiphase flow and geomechanics of faulted reservoirs by modeling faults as surfaces embedded in a three-dimensional medium by using zero-thickness interface elements.
Journal ArticleDOI
Backward SDE representation for stochastic control problems with nondominated controlled intensity.
Sébastien Choukroun,Andrea Cosso +1 more
TL;DR: In this article, the authors introduce a class of backward stochastic differential equations with jumps and partially constrained diffusive part, for which the minimal solution to their BSDE provides the unique viscosity solution to our fully nonlinear integro-partial differential equation.
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Inhomogeneous Hopf–Oleĭnik Lemma and regularity of semiconvex supersolutions via new barriers for the Pucci extremal operators
TL;DR: In this article, Caffarelli, Kohn, Spruck and Nirenberg constructed new barriers for the Pucci extremal operators with unbounded RHS and showed that these barriers can be obtained by a Harnack inequality up to the boundary type estimate.
Posted Content
Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces
TL;DR: The class of location-scale finite mixtures is of enduring interest both from applied and theoretical perspectives of probability and statistics as mentioned in this paper, and it has been shown that location scale mixtures of a continuous probability density function (PDF) can approximate any continuous PDF, uniformly, on a compact set.
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.
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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.