scholarly journals Exponential Dichotomies for Elliptic PDE on Radial Domains

2020 ◽  
pp. 49-68
Author(s):  
Margaret Beck ◽  
Graham Cox ◽  
Christopher Jones ◽  
Yuri Latushkin ◽  
Alim Sukhtayev
Keyword(s):  
Elliptic Pde ◽  
Exponential Dichotomies

An Accelerated Method for Nonlinear Elliptic PDE

2016 ◽  
Vol 69 (2) ◽  
pp. 556-580 ◽  
Author(s):  
Hayden Schaeffer ◽  
Thomas Y. Hou
Keyword(s):  
Elliptic Pde ◽  
Nonlinear Elliptic Pde ◽  
Nonlinear Elliptic

scholarly journals Nonuniform Exponential Dichotomies in Terms of Lyapunov Functions for Noninvertible Linear Discrete-Time Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ioan-Lucian Popa ◽  
Mihail Megan ◽  
Traian Ceauşu
Keyword(s):  
Discrete Time ◽  
Lyapunov Functions ◽  
Exponential Dichotomies ◽  
Discrete Time Systems ◽  
Nonuniform Exponential Dichotomies ◽  
Time Systems

The aim of this paper is to give characterizations in terms of Lyapunov functions for nonuniform exponential dichotomies of nonautonomous and noninvertible discrete-time systems.

scholarly journals Radial symmetry for an elliptic PDE with a free boundary

2021 ◽  
Vol 8 (26) ◽  
pp. 311-319
Author(s):  
Layan El Hajj ◽  
Henrik Shahgholian
Keyword(s):  
Free Boundary ◽  
Radial Symmetry ◽  
Regularity Of Solutions ◽  
Elliptic Pde ◽  
Linear Elliptic Equation ◽  
Boundary Harnack Principle ◽  
Partial Differential ◽  
Content Type ◽  
Moving Plane ◽  
Plane Technique

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u )  in  B 1 , 0 ≤ u > M ,  in  B 1 , u = M ,  on  ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

scholarly journals The quantum query complexity of elliptic PDE

2006 ◽  
Vol 22 (5) ◽  
pp. 691-725 ◽  
Author(s):  
Stefan Heinrich
Keyword(s):  
Query Complexity ◽  
Elliptic Pde ◽  
Quantum Query Complexity ◽  
Quantum Query

On Fredholm properties of Lu = u′ − A(t)u for paths of sectorial operators

2005 ◽  
Vol 135 (1) ◽  
pp. 39-60 ◽  
Author(s):  
Davide di Giorgio ◽  
Alessandra Lunardi
Keyword(s):  
Banach Space ◽  
Explicit Formula ◽  
Fredholm Operator ◽  
Finite Dimension ◽  
Sufficient Conditions ◽  
Spectral Flow ◽  
Exponential Dichotomies ◽  
Sectorial Operators ◽  
Common Domain ◽  
General Banach Space

We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.

Tempered exponential dichotomies: admissibility and stability under perturbations

2016 ◽  
Vol 31 (4) ◽  
pp. 525-545 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Exponential Dichotomies ◽  
Stability Under Perturbations

Conjugacies between linear and nonlinear non-uniform contractions

2008 ◽  
Vol 28 (1) ◽  
pp. 1-19 ◽  
Author(s):  
LUIS BARREIRA ◽  
CLAUDIA VALLS
Keyword(s):  
Banach Spaces ◽  
Discrete Time ◽  
Linear Operators ◽  
Nonlinear Perturbations ◽  
Exponential Dichotomies ◽  
Locally Lipschitz ◽  
Linear And Nonlinear

AbstractWe construct conjugacies between linear and nonlinear non-uniform exponential contractions with discrete time. We also consider the general case of a non-autonomous dynamics defined by a sequence of maps. The results are obtained by considering both linear and nonlinear perturbations of the dynamics xm+1=Amxm defined by a sequence of linear operators Am. In the case of conjugacies between linear contractions we describe them explicitly. All the conjugacies are locally Hölder, and in fact are locally Lipschitz outside the origin. We also construct conjugacies between linear and nonlinear non-uniform exponential dichotomies, building on the arguments for contractions. All the results are obtained in Banach spaces.

Sign in / Sign up

Export Citation Format

Share Document