scholarly journals Quantitative unique continuation for the heat equation with Coulomb potentials

2018 ◽  
Vol 8 (3) ◽  
pp. 1097-1116 ◽  
Author(s):  
Can Zhang ◽  
◽  
Keyword(s):  
Heat Equation ◽  
Unique Continuation ◽  
Quantitative Unique Continuation ◽  
Coulomb Potentials

scholarly journals Sets of Unique Continuation for Heat Equation

2014 ◽  
Vol 41 (4) ◽  
pp. 1267-1272
Author(s):  
Nikolai Nadirashvili ◽  
Nadezda Varkentina
Keyword(s):  
Heat Equation ◽  
Unique Continuation

scholarly journals Insensitizing control for linear and semi-linear heat equations with partially unknown domain

2019 ◽  
Vol 25 ◽  
pp. 50 ◽  
Author(s):  
Pierre Lissy ◽  
Yannick Privat ◽  
Yacouba Simporé
Keyword(s):  
Heat Equation ◽  
Control Problem ◽  
Duality Theory ◽  
Unique Continuation ◽  
Dirichlet Boundary Conditions ◽  
Linear Case ◽  
Heat Equations ◽  
Geometrical Approach ◽  
Linear Heat Equation ◽  
Linear Heat

We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝN (N ∈ ℕ*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.

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.

Sign in / Sign up

Export Citation Format

Share Document