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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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Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States
Pavol Quittner,Philippe Souplet +1 more
TL;DR: In this article, the authors propose a model for solving the model elliptic problems and model parabolic problems. But their model is based on Equations with Gradient Terms (EGS).
Book ChapterDOI
Multi-index Mittag-Leffler Functions
TL;DR: In this paper, Dzherbashian [Dzh60] defined a function with positive α 1 > 0, α 2 > 0 and real α 1, β 2, β 3, β 4, β 5, β 6, β 7, β 8, β 9, β 10, β 11, β 12, β 13, β 14, β 15, β 16, β 17, β 18, β 20, β 21, β 22, β 24
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Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p -Laplacian in $${\mathbb {R}}^N$$ R N
TL;DR: In this article, the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrodinger-Kirchhoff type was investigated, and multiplicity results were obtained by using the Ekeland variational principle and the Mountain Pass theorem.
References
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Book ChapterDOI
On the radial projection in normed spaces
TL;DR: In this article, the radial projection mapping of a real normed space with norm is defined, and the radial mapping is defined in terms of the radial norm of the projection space.
Journal ArticleDOI
Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications
TL;DR: In this paper, a maximum principle was proved for weak solutions of the telegraph equation in space dimension three and a counterexample for the maximum principle in dimension four was given.
Book ChapterDOI
Topics in Parabolic Equations: Some Typical Free Boundary Problems
TL;DR: In this article, it was shown that the Stefan problem is closely related to the theory of variational inequalities, and that the free boundary problem for the heat equation arising in the ice melting phenomenon is a special case of this problem.
Journal ArticleDOI
Self-Adjoint Operators and Cones
TL;DR: In this paper, it was shown that k A k n is an eigenvalue of A n and k A n has an eigvector in K. The role of the support-boundary of K in the simplicity of the principal eigen value kAk is investigated.
Journal ArticleDOI
Interior estimates for second-order elliptic differential (or finite-difference) equations via the maximum principle
TL;DR: In this paper, second-order elliptic operators are transformed into secondorder operators of a higher dimensionality acting on differences of functions and applied the maximum principle to the resulting operators yields various a-priori pointwise estimates to difference-quotients of solutions of elliptic differential, as well as finite difference, equations.