scholarly journals A note on immersions of domains of fractional powers of certain sectorial operators in Sobolev spaces

2012 ◽  
Vol 25 (12) ◽  
pp. 2105-2109 ◽  
Author(s):  
Anderson Luis A. de Araujo ◽  
José Luiz Boldrini
Keyword(s):  
Sobolev Spaces ◽  
Fractional Powers ◽  
Sectorial Operators

scholarly journals The Global Existence of Nonlinear Evolutionary Equation with Small Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Xunwu Yin
Keyword(s):  
Global Existence ◽  
Local Existence ◽  
Evolutionary Equation ◽  
Fractional Powers ◽  
Gronwall’S Inequality ◽  
Sectorial Operators ◽  
Local Existence And Uniqueness ◽  
Small Delay ◽  
Fixed Point Principle ◽  
Nonlinear Evolutionary Equation

We investigate the global existence of the delayed nonlinear evolutionary equation∂tu+Au=f(u(t),u(t−τ)). Our work space is the fractional powers spaceXα. Under the fundamental theorem on sectorial operators, we make use of the fixed-point principle to prove the local existence and uniqueness theorem. Then, the global existence is obtained by Gronwall’s inequality.

scholarly journals On the harmonic extension approach to fractional powers in Banach spaces

2020 ◽  
Vol 23 (4) ◽  
pp. 1054-1089
Author(s):  
Jan Meichsner ◽  
Christian Seifert
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Banach Spaces ◽  
Incomplete Data ◽  
Existence And Uniqueness ◽  
Bounded Solution ◽  
Fractional Powers ◽  
Order Ordinary Differential Equation ◽  
Harmonic Extension ◽  
Sectorial Operators

AbstractWe show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e., of the harmonic extension).

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

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