scholarly journals On two methods for quantitative unique continuation results for some nonlocal operators

2020 ◽  
Vol 45 (11) ◽  
pp. 1512-1560
Author(s):  
María Ángeles García-Ferrero ◽  
Angkana Rüland
Keyword(s):  
Unique Continuation ◽  
Nonlocal Operators ◽  
Quantitative Unique Continuation

scholarly journals Band Edge Localization Beyond Regular Floquet Eigenvalues

2020 ◽  
Vol 21 (7) ◽  
pp. 2151-2166
Author(s):  
Albrecht Seelmann ◽  
Matthias Täufer
Keyword(s):  
Floquet Theory ◽  
Large Deviation ◽  
Random Variables ◽  
Unique Continuation ◽  
Band Edge ◽  
Random Schrödinger Operators ◽  
Initial Scale ◽  
Quantitative Unique Continuation ◽  
Scale Estimate ◽  
Floquet Eigenvalues

Abstract We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $$L^2({\mathbb {R}}^d)$$ L 2 ( R d ) is universal. By this, we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.

The bang-bang property of time and norm optimal control problems for parabolic equations with time-varying fractional Laplacian

2019 ◽  
Vol 25 ◽  
pp. 7 ◽  
Author(s):  
Xin Yu ◽  
Liang Zhang
Keyword(s):  
Optimal Control ◽  
Bounded Domain ◽  
Parabolic Equations ◽  
Fractional Laplacian ◽  
Optimal Control Problems ◽  
Unique Continuation ◽  
Control Problems ◽  
Time Varying ◽  
Observability Inequality ◽  
Quantitative Unique Continuation

In this paper, we establish the bang-bang property of time and norm optimal control problems for parabolic equations governed by time-varying fractional Laplacian, evolved in a bounded domain of ℝd. We firstly get a quantitative unique continuation at one point in time for parabolic equations governed by time-varying fractional Laplacian. Then, we establish an observability inequality from measurable sets in time for solutions of the above-mentioned equations. Finally, with the aid of the observability inequality, the bang-bang property of time and norm optimal control problems can be obtained.

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