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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
Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain
TL;DR: In this article, the initial boundary value problem of two dimensional MHD equations without magnetic diffusion in a strip domain was studied and it was proved that the MHD equation has a unique global strong solution around the equilibrium state for both the non-slip boundary condition and Navier slip boundary condition on the velocity.
Journal ArticleDOI
Stabilization of a Thermoelastic Laminated Beam with Past History
Wenjun Liu,Weifan Zhao +1 more
TL;DR: In this paper, the well-posedness and asymptotic stability of a thermoelastic laminated beam with past history was studied. And the authors proved the exponential and polynomial stability of the system with and without structural damping.
Journal ArticleDOI
Wellposedness and Decay Rates for the Cauchy Problem of the Moore–Gibson–Thompson Equation Arising in High Intensity Ultrasound
TL;DR: In this paper, the Moore-Gibson-Thompson equation was studied in the frequency domain by analyzing the eigenvalues of the Fourier image of the solution and writing the solution accordingly.
Proceedings ArticleDOI
GuSTO: Guaranteed Sequential Trajectory optimization via Sequential Convex Programming
TL;DR: GuSTO as discussed by the authors is an algorithmic framework to solve trajectory optimization problems for control-affine systems with drift and enjoys theoretical convergence guarantees in terms of convergence to, at least, a stationary point.
Journal ArticleDOI
An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise
Viorel Barbu,Michael Röckner +1 more
TL;DR: In this article, a general approach to the existence and uniqueness theory of infinite dimensional stochastic equations of the form dX(t) +A(t, X(t))dt = X( t)dW (t) in (0, T ) × H, where A is a time-dependent nonlinear monotone and demicontinuous operator from V to V I, coercive and with poly-nomial growth.
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.