The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $

2001 ◽  
Vol 1 (4) ◽  
pp. 361-385 ◽  
Author(s):  
Pascal Auscher ◽  
Steve Hofmann ◽  
Alan McIntosh ◽  
Philippe Tchamitchian
Keyword(s):  
Higher Order ◽  
Elliptic Operators ◽  
Square Root ◽  
Kato Square Root Problem

The Solution of the Kato Square Root Problem for Second Order Elliptic Operators on \Bbb R n

2002 ◽  
Vol 156 (2) ◽  
pp. 633 ◽  
Author(s):  
Pascal Auscher ◽  
Steve Hofmann ◽  
Michael Lacey ◽  
Alan McIntosh ◽  
Ph. Tchamitchian
Keyword(s):  
Elliptic Operators ◽  
Second Order ◽  
Square Root ◽  
Second Order Elliptic Operators ◽  
Kato Square Root Problem

scholarly journals HIGHER-ORDER CURVATURE TERMS IN BORN–INFELD TYPE BRANE THEORIES

2011 ◽  
Vol 20 (01) ◽  
pp. 59-75 ◽  
Author(s):  
EFRAIN ROJAS
Keyword(s):  
Ricci Tensor ◽  
General Structure ◽  
Extrinsic Curvature ◽  
Higher Order ◽  
Square Root ◽  
Auxiliary Variables ◽  
Field Equations ◽  
Alternative Description ◽  
Brane Models ◽  
Combined Matrix

The field equations associated to Born–Infeld type brane theories are studied by using auxiliary variables. This approach hinges on the fact, that the expressions defining the physical and geometrical quantities describing the worldvolume are varied independently. The general structure of the Born–Infeld type theories for branes contains the square root of a determinant of a combined matrix between the induced metric on the worldvolume swept out by the brane and a symmetric/antisymmetric tensor depending on gauge, matter or extrinsic curvature terms taking place on the worldvolume. The higher-order curvature terms appearing in the determinant form come to play in competition with other effective brane models. Additionally, we suggest a Born–Infeld–Einstein type action for branes where the higher-order curvature content is provided by the worldvolume Ricci tensor. This action provides an alternative description of the dynamics of braneworld scenarios.

scholarly journals ELLIPTIC OPERATORS OF HIGHER ORDER IN DIVERGENCE FORM

1968 ◽  
Vol 59 (4) ◽  
pp. 1067-1067
Author(s):  
U. Neri
Keyword(s):  
Divergence Form ◽  
Higher Order ◽  
Elliptic Operators

Square-root-like higher-order topological states in three-dimensional sonic crystals

2021 ◽  
Author(s):  
Zhiguo Geng ◽  
Huanzhao Lv ◽  
Zhan Xiong ◽  
Yu-Gui Peng ◽  
Zhaojiang Chen ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Surface States ◽  
Higher Order ◽  
Square Root ◽  
Root Lattice ◽  
Third Order ◽  
Topological Materials ◽  
Topological States ◽  
Sonic Crystal ◽  
Sonic Crystals

Abstract The square-root descendants of higher-order topological insulators were proposed recently, whose topological property is inherited from the squared Hamiltonian. Here we present a three-dimensional (3D) square-root-like sonic crystal by stacking the 2D square-root lattice in the normal (z) direction. With the nontrivial intralayer couplings, the opened degeneracy at the K-H direction induces the emergence of multiple acoustic localized modes, i.e., the extended 2D surface states and 1D hinge states, which originate from the square-root nature of the system. The square-root-like higher order topological states can be tunable and designed by optionally removing the cavities at the boundaries. We further propose a third-order topological corner state in the 3D sonic crystal by introducing the staggered interlayer couplings on each square-root layer, which leads to a nontrivial bulk polarization in the z direction. Our work sheds light on the high-dimensional square-root topological materials, and have the potentials in designing advanced functional devices with sound trapping and acoustic sensing.

scholarly journals THE KATO SQUARE ROOT PROBLEM FOR MIXED BOUNDARY VALUE PROBLEMS

2006 ◽  
Vol 74 (01) ◽  
pp. 113-130 ◽  
Author(s):  
ANDREAS AXELSSON ◽  
STEPHEN KEITH ◽  
ALAN McINTOSH
Keyword(s):  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Square Root ◽  
Mixed Boundary Value Problems ◽  
Kato Square Root Problem

scholarly journals Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations

2020 ◽  
Author(s):  
Ariel Barton ◽  
Steve Hofmann ◽  
Svitlana Mayboroda
Keyword(s):  
Neumann Problem ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Higher Order ◽  
Elliptic Operators ◽  
Layer Potentials ◽  
The Neumann Problem

Abstract We solve the Neumann problem, with nontangential estimates, for higher-order divergence form elliptic operators with variable $t$-independent coefficients. Our results are accompanied by nontangential estimates on higher-order layer potentials.

scholarly journals The Kato square root problem on submanifolds

2012 ◽  
Vol 86 (3) ◽  
pp. 879-910 ◽  
Author(s):  
Andrew J. Morris
Keyword(s):  
Square Root ◽  
Kato Square Root Problem

Pointwise lower bounds on the heat kernels of higher order elliptic operators

1999 ◽  
Vol 125 (1) ◽  
pp. 105-111 ◽  
Author(s):  
E. B. DAVIES
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Space Dimension ◽  
Adjoint Operator ◽  
Integral Kernel ◽  
Divergence Form ◽  
Higher Order ◽  
Elliptic Operators ◽  
Heat Kernels ◽  
Second Order Elliptic Operators

Suppose that H=H*[ges ]0 on L2(X, dx) and that e−Ht has an integral kernel K(t, x, y) which is a continuous function of all three variables. It follows from the fact that e−Ht is a non-negative self-adjoint operator that K(t, x, x)[ges ]0 for all t>0 and x∈X. Our main abstract results, Theorems 2 and 3, provide a positive lower bound on K(t, x, x) under suitable general hypotheses. As an application we obtain a explicit positive lower bound on K(t, x, y) when x is close enough to y and H is a higher order uniformly elliptic operator in divergence form acting in L2(RN, dx); see Theorem 6.We emphasize that our results are not applicable to second order elliptic operators (except in one space dimension). For such operators much stronger lower bounds can be obtained by an application of the Harnack inequality. For higher order operators, however, we believe that our result is the first of its type which does not impose any continuity conditions on the highest order coefficients of the operators.

On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

1998 ◽  
Vol 57 (3) ◽  
pp. 305-314
Author(s):  
El Maati Ouhabaz
Keyword(s):  
Spectral Function ◽  
Higher Order ◽  
Elliptic Operators ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic
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