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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
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NLS ground states on graphs
TL;DR: In this paper, the existence of ground states for the subcritical NLS energy on metric graphs was investigated and a new rearrangement technique adapted to the graph where it applies was introduced.
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Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics
TL;DR: In this article, the authors analyze under which conditions equilibration between two competing effects, repulsion and attraction, occurs, and show the existence of global minimizers among these equilibria.
Journal ArticleDOI
Analysis of degenerate cross-diffusion population models with volume filling
Nicola Zamponi,Ansgar Jüngel +1 more
TL;DR: In this paper, a class of parabolic cross-diffusion systems modeling the interaction of an arbitrary number of population species is analyzed in a bounded domain with no-flux boundary conditions.
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On the optimality of shape and data representation in the spectral domain
TL;DR: A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in repre- senting smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis, and a numerical acceleration technique for classical scaling is reviewed.
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On the existence of stationary solutions for higher-order p-Kirchhoff problems
TL;DR: In this paper, Colasuonno et al. established the existence of two nontrivial weak solutions of possibly degenerate nonlinear eigenvalue problems involving the p-polyharmonic Kirchhoff operator in bounded domains.
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.