Open AccessBook
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
More filters
The Atiyah-Singer index theorem
Nigel Higson,John Roe +1 more
TL;DR: The Atiyah-Singer Index Theorem as mentioned in this paper provides a topological formula, in terms of characteristic classes, of the Fredholm index of certain elliptic operators, and had influenced, since its discovery in the 1960s, many areas of mathematics.
Journal ArticleDOI
Incompressible immiscible multiphase flows in porous media: a variational approach
TL;DR: In this paper, the competitive motion of (N + 1) incompressible immiscible phases within a porous medium is described as the gradient flow of a singular energy in the space of non-negative measures with prescribed mass endowed with some tensorial Wasserstein distance.
Journal ArticleDOI
Solving and learning nonlinear PDEs with Gaussian processes
TL;DR: In this article, a generalization of collocation kernel methods to nonlinear partial differential equations (PDEs) and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes is proposed.
Journal ArticleDOI
Output feedback stabilization of the Korteweg–de Vries equation
Swann Marx,Eduardo Cerpa +1 more
TL;DR: The local exponential stability of the closed-loop system is proven and the output feedback control law for the Korteweg-de Vries equation is presented, based on the backstepping method and the introduction of an appropriate observer.
Journal ArticleDOI
Sliding mode control of Schrödinger equation-ODE in the presence of unmatched disturbances
Wen Kang,Wen Kang,Emilia Fridman +2 more
TL;DR: The backstepping method is first applied to transform the system into an equivalent target system where the target system is input-to-state stable, and the sliding mode control (SMC) law is designed for thetarget system to reject the matched disturbance.
References
More filters
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.