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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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Weak form of Stokes-Dirac structures and geometric discretization of port-Hamiltonian systems
TL;DR: The mixed Galerkin discretization of distributed-parameter port-Hamiltonian systems is presented and its geometric approximation by a finite-dimensional Dirac structure and power-preserving maps on the space of discrete power variables are derived.
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Boundary layer analysis of the Navier–Stokes equations with generalized Navier boundary conditions
Gung-Min Gie,James P. Kelliher +1 more
TL;DR: In this article, the weak boundary layer phenomenon of the Navier-Stokes equations with generalized Navier friction boundary conditions, u⋅n = 0, [S(u)n]tan+Au=0, in a bounded domain in R3 when the viscosity, e>0, is studied.
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A new duality approach to elasticity
TL;DR: In this paper, the displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field or the stress tensor field is the unknown.
Journal ArticleDOI
Gradient discretization of hybrid dimensional Darcy flows in fractured porous media
TL;DR: In this article, the convergence analysis is carried out in the framework of gradient schemes which accounts for a large family of conforming and nonconforming discretizations of hybrid dimensional Darcy flows in fractured porous media.
Journal ArticleDOI
Semilinear elliptic equations involving mixed local and nonlocal operators
TL;DR: In this paper, an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type is considered, and the symmetry properties of the solutions are investigated.
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.