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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 Ulam–Hyers stability of first order differential systems with nonlocal initial conditions
Sz. András,J.J. Kolumbán +1 more
TL;DR: In this article, the stability of first order differential systems with nonlocal initial conditions on compact intervals and on non-compact intervals with a suitable weight function w was investigated, using vectorial norms, convergent to zero matrices, Sobolev spaces.
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Sobolev spaces of symmetric functions and applications
TL;DR: In this paper, the authors obtained sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball and obtained some imbeddings of such spaces in weighted L q -spaces.
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Shape Optimization of a Coupled Thermal Fluid-Structure Problem in a Level Set Mesh Evolution Framework
TL;DR: Hadamard's method of shape differentiation is applied to topology optimization of a weakly coupled three physics problem in this article, where shape sensitivities are derived in a fully Lagrangian setting which allows us to obtain shape derivatives of general objective functions.
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Localization Analysis of an Energy-Based Fourth-Order Gradient Plasticity Model
TL;DR: In this article, an energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto-plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening.
Journal ArticleDOI
Shape deformation analysis from the optimal control viewpoint
TL;DR: In this paper, a general approach to shape deformation analysis, within the framework of optimal control theory, was proposed, in which a deformation is represented as the flow of diffeomorphisms generated by time-dependent vector fields.
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.