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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
On the C-property and w∗-representations of risk measures
Niushan Gao,Foivos Xanthos +1 more
TL;DR: In this paper, the authors identify a large class of Orlicz spaces LΦ(μ) for which the topology σ(L Φ (μ),LΦ (m)n∼) fails the C-property introduced by Biagini and Frittelli and use it to prove a w∗representation theorem for proper convex increasing functionals.
Posted Content
Convergence of tamed Euler schemes for a class of stochastic evolution equations
TL;DR: In this paper, the stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term were proved.
Journal ArticleDOI
Infinite-Horizon Bilinear Optimal Control Problems: Sensitivity Analysis and Polynomial Feedback Laws
TL;DR: An infinite-horizon optimal control problem subject to an infinite-dimensional state equation with state and control variables appearing in a bilinear form is investigated.
Journal ArticleDOI
Analytical basis for determining slope lines in grid digital elevation models
TL;DR: In this article, an analytical basis for the determination of slope lines in grid digital elevation models is provided by using the D8-LTD method (eight slope directions, least transverse deviation).
Journal ArticleDOI
Controllability of the Heat Equation with an Inverse-Square Potential Localized on the Boundary
TL;DR: The previous results showed that for any value of parameters $\mu\leq \mu(N):=(N-2)^2/4$, the corresponding parabolic system can be controlled to zero with the control distributed in any open subset of the domain.
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.