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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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Quasilinear elliptic equations and weighted Sobolev-Poincaré inequalities with distributional weights
TL;DR: In this article, a class of weak solutions to the quasilinear Schrodinger operator was introduced, tailored to general distributional coefficients σ which satisfy the inequality − Λ ∫ Ω | ∇ h | p d x ≤ 〈 | h| p, σ 〉 ≤ λ ∫ ǫ | | h ∈ C 0 ∞ ( Ω ).
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Boundary regularity and sufficient conditions for strong local minimizers
TL;DR: In this paper, a new proof of the sufficiency theorem for strong local minimizers concerning C 1 -extremals at which the second variation is strictly positive is presented in the quasiconvex setting, in accordance with the original statement by Grabovsky and Mengesha.
Posted Content
On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics
Weinan E,Stephan Wojtowytsch +1 more
TL;DR: In this article, the authors define the Banach spaces for ReLU neural networks of finite depth $L$ and infinite width and show that these spaces can be approximated by multi-layer neural networks with dimension-independent convergence rates.
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Hamiltonian treatment of linear field theories in the presence of boundaries: a geometric approach
TL;DR: In this article, the authors studied the constraint structure of the Hamiltonian description for the scalar and electromagnetic fields in the presence of spatial boundaries and carefully discussed the implementation of the geometric constraint algorithm of Gotay, Nester and Hinds.
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Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres
TL;DR: In this paper, the authors studied the behavior of the iterated Douglas-Rachford method for a line and a circle by considering two generalizations: the line and an ellipse and that of a line together with a p-sphere.
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.