scholarly journals Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems

2011 ◽  
Vol 250 (9) ◽  
pp. 3630-3651 ◽  
Author(s):  
Nenad Antonić ◽  
Krešimir Burazin
Keyword(s):  
Boundary Conditions ◽  
Boundary Operator ◽  
Friedrichs Systems

Absorbing boundary conditions for acoustic media

Geophysics ◽  
1985 ◽  
Vol 50 (6) ◽  
pp. 892-902 ◽  
Author(s):  
R. G. Keys
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Plane Waves ◽  
Basic Element ◽  
Absorbing Boundary Conditions ◽  
Boundary Operator ◽  
Order Differential Operator ◽  
Absorbing Boundary ◽  
Boundary Operators

By decomposing the acoustic wave equation into incoming and outgoing components, an absorbing boundary condition can be derived to eliminate reflections from plane waves according to their direction of propagation. This boundary condition is characterized by a first‐order differential operator. The differential operator, or absorbing boundary operator, is the basic element from which more complicated boundary conditions can be constructed. The absorbing boundary operator can be designed to absorb perfectly plane waves traveling in any two directions. By combining two or more absorption operators, boundary conditions can be created which absorb plane waves propagating in any number of directions. Absorbing boundary operators simplify the task of designing boundary conditions to reduce the detrimental effects of outgoing waves in many wave propagation problems.

scholarly journals Boundary RG flows for fermions and the mod 2 anomaly

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Philip Boyle Smith ◽  
David Tong
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Degrees Of Freedom ◽  
Central Charge ◽  
Boundary Operator ◽  
Majorana Fermions ◽  
Boundary State ◽  
Infra Red ◽  
Boundary Operators ◽  
Relevant Operator

Boundary conditions for Majorana fermions in d=1+1d=1+1 dimensions fall into one of two SPT phases, associated to a mod 2 anomaly. Here we consider boundary conditions for 2N2N Majorana fermions that preserve a U(1)^NU(1)N symmetry. In general, the left-moving and right-moving fermions carry different charges under this symmetry, and implementation of the boundary condition requires new degrees of freedom, which manifest themselves in a boundary central charge gg. We follow the boundary RG flow induced by turning on relevant boundary operators. We identify the infra-red boundary state. In many cases, the boundary state flips SPT class, resulting in an emergent Majorana mode needed to cancel the anomaly. We show that the ratio of UV and IR boundary central charges is given by g^2_{IR} / g^2_{UV} = \mathrm{dim} \, \mathcal{O}gIR2/gUV2=dim𝒪, the dimension of the perturbing boundary operator. Any relevant operator necessarily has \mathrm{dim} \, \mathcal{O} < 1dim𝒪<1, ensuring that the central charge decreases in accord with the gg-theorem.

COMPACTNESS RESULTS FOR TRANSPORT EQUATIONS AND APPLICATIONS

2001 ◽  
Vol 11 (07) ◽  
pp. 1181-1202 ◽  
Author(s):  
KHALID LATRACH
Keyword(s):  
Boundary Conditions ◽  
Collision Operator ◽  
Transport Equations ◽  
Boundary Operator ◽  
Systematic Analysis ◽  
Boundary Operators ◽  
Variable Space ◽  
Abstract Boundary ◽  
Two Parameters ◽  
Transport Problems

The goal of this paper is to give a systematic analysis of compactness properties for transport equations with general boundary conditions where an abstract boundary operator relates the incoming and outgoing fluxes. The analysis involves two parameters: The velocity measure and the collision operator. Hence, for a large class of (velocity) measures and under appropriate assumptions on scattering operators compactness results are obtained. Using the positivity (in the lattice sense) and the comparison arguments by Dodds–Fremlin, their converses are derived, and necessary conditions for some remainder term of the Dyson–Phillips expansion to be compact are given. Our results are independent of the properties of the boundary operators and play a crucial role in the understanding of the time asymptotic structure of evolution transport problems. Also, although solutions of transport equations propagate singularities (due to the hyperbolic nature of the operator), they bring the regularity in the variable space (regardless of the boundary operator). We end the paper by applying the obtained results to discuss the existence of solutions to a nonlinear boundary value problem and to describe in detail the various essential spectra of transport operators with abstract boundary conditions.

On BVPS for Strongly Elliptic Systems with Higher Order Boundary Conditions

2007 ◽  
Vol 14 (1) ◽  
pp. 145-167
Author(s):  
Flavia Lanzara
Keyword(s):  
Boundary Conditions ◽  
Explicit Formula ◽  
Singular Integral ◽  
Elliptic Systems ◽  
Higher Order ◽  
Boundary Operator ◽  
Compatibility Conditions ◽  
Layer Potential ◽  
Integral System ◽  
Regular Type

Abstract We consider BVPs for strongly elliptic systems of order 2𝑙 with the boundary conditions of order 𝑙 + 𝑛, 𝑛 ⩾ 0. By representing the solution by means of a simple layer potential of order 𝑛 and by imposing the boundary conditions, we get a singular integral system which is of regular type if and only if the boundary operator satisfies the Lopatinskiĭ condition and which can be solved if suitable compatibility conditions are satisfied. An explicit formula for computing the index of the BVP is given.

Expansion theorems for Birkhoff-regular differential-boundary operators

1987 ◽  
Vol 107 (3-4) ◽  
pp. 349-374 ◽  
Author(s):  
Manfred Möller
Keyword(s):  
Boundary Conditions ◽  
Integral Condition ◽  
Boundary Operator ◽  
Boundary Part ◽  
Associated Functions ◽  
Multipoint Boundary Conditions ◽  
Boundary Operators ◽  
Birkhoff Regularity ◽  
Regular Boundary Conditions ◽  
Differential Part

SynopsisIn this paper we consider differential-boundary operators T over a finite interval depending on a complex parameter. A differential-boundary operator admits boundary conditions in the differential part. The boundary part contains multipoint boundary conditions and integral conditions. For Birkhoff-regular boundary conditions we prove that every Lp -function is expansible into a series with respect to the eigenfunctions and the associated functions of the differential-boundary operator. Here the Birkhoff-regularity only depends on the boundary conditions at the endpoints of the interval, i.e. T is Birkhoff-regular if and only if T0 is Birkhoff-regular where T0 arises from T by omitting the boundary part in the differential equations, the interior point boundary conditions and the integral condition.

scholarly journals On a Boundary Value Problem for the Biharmonic Equation with Multiple Involutions

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2020
Author(s):  
Batirkhan Turmetov ◽  
Valery Karachik ◽  
Moldir Muratbekova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Boundary Value ◽  
Boundary Operator ◽  
Biharmonic Operator ◽  
Uniqueness Of Solutions ◽  
Neumann Type ◽  
Type Boundary

A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution type were considered. The properties of the matrices of these transformations were investigated. As applications of the considered transformations, the questions about the solvability of a boundary value problem for a nonlocal biharmonic equation were studied. Modified Hadamard derivatives were considered as the boundary operator. The considered problems covered the Dirichlet and Neumann-type boundary conditions. Theorems on the existence and uniqueness of solutions to the studied problems were proven.

Intrinsic Boundary Conditions for Friedrichs Systems

2010 ◽  
Vol 35 (9) ◽  
pp. 1690-1715 ◽  
Author(s):  
Nenad Antonić ◽  
Krešimir Burazin
Keyword(s):  
Boundary Conditions ◽  
Friedrichs Systems

scholarly journals NONLOCALITY AND ELLIPTICITY IN A GAUGE-INVARIANT QUANTIZATION

2000 ◽  
Vol 15 (03) ◽  
pp. 449-460 ◽  
Author(s):  
GIAMPIERO ESPOSITO ◽  
COSIMO STORNAIOLO
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Free Particle ◽  
Nonlocal Boundary ◽  
Mixed Boundary Conditions ◽  
Nonlocal Boundary Conditions ◽  
Two Dimensions ◽  
Boundary Operator ◽  
Positive Order ◽  
Gauge Invariant

The quantum theory of a free particle in two dimensions with nonlocal boundary conditions on a circle is known to lead to surface and bulk states. Such a scheme is here generalized to the quantized Maxwell field, subject to mixed boundary conditions. If the Robin sector is modified by the addition of a pseudo-differential boundary operator, gauge-invariant boundary conditions are obtained at the price of dealing with gauge-field and ghost operators which become pseudo-differential. A good elliptic theory is then obtained if the kernel occurring in the boundary operator obeys certain summability conditions, and it leads to a peculiar form of the asymptotic expansion of the symbol. The cases of ghost operator of negative and positive order are studied within this framework.

scholarly journals On analytic bootstrap for interface and boundary CFT

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Parijat Dey ◽  
Alexander Söderberg
Keyword(s):  
Boundary Conditions ◽  
Neumann Boundary ◽  
Equation Of Motion ◽  
Neumann Boundary Conditions ◽  
Boundary Operator ◽  
Analytic Structure ◽  
Conformal Boundary ◽  
Conformal Blocks ◽  
Operator Expansion ◽  
Bootstrap Techniques

Abstract We use analytic bootstrap techniques for a CFT with an interface or a boundary. Exploiting the analytic structure of the bulk and boundary conformal blocks we extract the CFT data. We further constrain the CFT data by applying the equation of motion to the boundary operator expansion. The method presented in this paper is general, and it is illustrated in the context of perturbative Wilson-Fisher theories. In particular, we find constraints on the OPE coefficients for the interface CFT in 4 − ϵ dimensions (upto order $$ \mathcal{O} $$ O (ϵ2)) with ϕ4-interactions in the bulk. We also compute the corresponding coefficients for the non-unitary ϕ3-theory in 6 − ϵ dimensions in the presence of a conformal boundary equipped with either Dirichlet or Neumann boundary conditions upto order $$ \mathcal{O} $$ O (ϵ), or an interface upto order $$ \mathcal{O}\left(\sqrt{\epsilon}\right) $$ O ϵ .

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