scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

scholarly journals $T\bar T$ and the mirage of a bulk cutoff

2021 ◽  
Vol 10 (2) ◽  
Author(s):  
Monica Guica ◽  
Ruben Monten
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Stress Tensor ◽  
Deformation Parameter ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Gravitational Theory ◽  
Dirichlet Boundary Conditions ◽  
Bulk Solution ◽  
Matter Fields

We use the variational principle approach to derive the large NN holographic dictionary for two-dimen-sional T\bar TTT‾-deformed CFTs, for both signs of the deformation parameter. The resulting dual gravitational theory has mixed boundary conditions for the non-dynamical graviton; the boundary conditions for matter fields are undeformed. When the matter fields are turned off and the deformation parameter is negative, the mixed boundary conditions for the metric at infinity can be reinterpreted on-shell as Dirichlet boundary conditions at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. The holographic stress tensor of the deformed CFT is fixed by the variational principle, and in pure gravity it coincides with the Brown-York stress tensor on the radial bulk slice with a particular cosmological constant counterterm contribution. In presence of matter fields, the connection between the mixed boundary conditions and the radial ``bulk cutoff’’ is lost. Only the former correctly reproduce the energy of the bulk configuration, as expected from the fact that a universal formula for the deformed energy can only depend on the universal asymptotics of the bulk solution, rather than the details of its interior. The asymptotic symmetry group associated with the mixed boundary conditions consists of two commuting copies of a state-dependent Virasoro algebra, with the same central extension as in the original CFT.

scholarly journals The Singular Perturbation Problem for a Class of Generalized Logistic Equations Under Non-classical Mixed Boundary Conditions

2019 ◽  
Vol 19 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Sergio Fernández-Rincón ◽  
Julián López-Gómez
Keyword(s):  
Singular Perturbation ◽  
Boundary Conditions ◽  
Distance Function ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Logistic Equations ◽  
Perturbation Problem ◽  
Second Order Elliptic Operators

Abstract This paper studies a singular perturbation result for a class of generalized diffusive logistic equations, {d\mathcal{L}u=uh(u,x)} , under non-classical mixed boundary conditions, {\mathcal{B}u=0} on {\partial\Omega} . Most of the precursors of this result dealt with Dirichlet boundary conditions and self-adjoint second order elliptic operators. To overcome the new technical difficulties originated by the generality of the new setting, we have characterized the regularity of {\partial\Omega} through the regularity of the associated conormal projections and conormal distances. This seems to be a new result of a huge relevance on its own. It actually complements some classical findings of Serrin, [39], Gilbarg and Trudinger, [21], Krantz and Parks, [27], Foote, [18] and Li and Nirenberg [28] concerning the regularity of the inner distance function to the boundary.

scholarly journals The condition number of the BEM-matrix arising from Laplace's equation

2007 ◽  
Vol 4 (2) ◽  
Author(s):  
W. Dijkstra ◽  
R.M.M. Mattheij
Keyword(s):  
Boundary Element Method ◽  
Boundary Conditions ◽  
Condition Number ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Circular Domain ◽  
Block Matrix ◽  
Dirichlet Boundary Conditions ◽  
System Matrix

We investigate the condition number of the matrices that appear in the boundary element method. In particular we consider the Laplace equation with mixed boundary conditions. For Dirichlet boundary conditions, the condition number of the system matrix increases linearly with the number of boundary elements. We extend the research and search for a relation between the condition number and the number of elements in the case of mixed boundary conditions. In the case of a circular domain, we derive an estimate for the condition number of the system matrix. This matrix consists of two blocks, each block originating from a well-conditioned matrix. We show that the block matrix is also well-conditioned.

scholarly journals Stability of Optimal Controls for the Stationary Boussinesq Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Gennady Alekseev ◽  
Dmitry Tereshko
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boussinesq Equations ◽  
Mixed Boundary Conditions ◽  
State Equation ◽  
Dirichlet Boundary Conditions ◽  
Heat Conducting ◽  
Optimal Controls ◽  
Viscous Heat ◽  
Cost Functionals

The stationary Boussinesq equations describing the heat transfer in the viscous heat-conducting fluid under inhomogeneous Dirichlet boundary conditions for velocity and mixed boundary conditions for temperature are considered. The optimal control problems for these equations with tracking-type functionals are formulated. A local stability of the concrete control problem solutions with respect to some disturbances of both cost functionals and state equation is proved.

scholarly journals Large-charge limit of AdS boson stars with mixed boundary conditions

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Shi-Fa Guo ◽  
Hai-Shan Liu ◽  
H. Lü ◽  
Yi Pang
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Numerical Data ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Massless Scalar ◽  
Boson Stars ◽  
Universal Structure ◽  
Close Form ◽  
Large Charge

Abstract It was recently shown that charged AdS boson stars can reproduce the universal structure of the lowest scaling dimension in the subsector of a CFT with fixed large global U(1) charge Q. Using the model consisting of Einstein-Maxwell gravity with a negative cosmological constant, coupled to a U(1)-charged conformally massless scalar with the fourth-order self interaction, we construct a class of charged AdS boson star solutions in the large Q limit, where the scalar field obeys a mixed boundary condition, parameterized by k that interpolates between the Neumann and Dirichlet boundary conditions corresponding to k = 0 and ∞ respectively. By varying k, we numerically read off the k dependence of the leading coefficient c3/2(k) ≡ limQ→∞M/Q3/2. We find that c3/2(k) is a monotonously increasing function which grows linearly when k is sufficiently small. When k → ∞, c3/2(k) approaches the maximal value at a decreasing rate given by k−3/2. We also obtain a close form expression that fits the numerical data for the entire range of k within 10−4 accuracy.

Effects of Pressure Boundary on Dynamic Torque Behavior of Hydroviscous Drive

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
Jianzhong Cui ◽  
Jun Liu ◽  
Fangwei Xie ◽  
Cuntang Wang ◽  
Pengliang Hou
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Pressure Profile ◽  
Neumann Boundary Conditions ◽  
Torque Control ◽  
Dirichlet Boundary Conditions ◽  
Squeeze Film ◽  
Soft Start

The object of this work is to investigate the effect of the change of film pressure resulting from axial squeeze-film motion between driving and driven disks on the performance of hydroviscous drive (HVD). A simplified mathematical model of the steady and laminar flow between parallel disks is established with consideration of three kinds of pressure boundary conditions. Some analytical solutions of film thickness, rotate speed of driven disk, viscous torque, and total torque are obtained. The numerical results show that the torque response depends on the relationship between the inlet pressure and the outlet pressure when considering the Dirichlet boundary conditions. The soft-start under Dirichlet boundary conditions and Mixed boundary conditions reflects the constant-torque startup and torque control startup, respectively. Compared with the two boundary conditions above, the soft-start under pressure profile boundary from Neumann boundary conditions has advantages for speed regulation. The effects of the ratio of inner and outer radius on the torque profiles and soft-start time are mainly related to Dirichlet boundary conditions and pressure profile boundary from Neumann boundary conditions.

scholarly journals Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term

2021 ◽  
Author(s):  
Jon Asier Bárcena-Petisco
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Space Variable ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Null Controllability ◽  
Dirichlet Boundary Conditions ◽  
The Cost ◽  
Transport Term

In this paper we consider the heat equation with Neumann, Robin and mixed boundary conditions (with coefficients on the boundary which depend on the space variable). The main results concern the behaviour of the cost of the null controllability with respect to the diffusivity when the control acts in the interior. First, we prove that if we almost have Dirichlet boundary conditions in the part of the boundary in which the flux of the transport enters, the cost of the controllability decays for a time $T$ sufficiently large. Next, we show some examples of Neumann and mixed boundary conditions in which for any time $T>0$ the cost explodes exponentially as the diffusivity vanishes. Finally, we study the cost of the problem with Neumann boundary conditions when the control is localized in the whole domain.

L∞-ESTIMATES FOR DIVERGENCE OPERATORS ON BAD DOMAINS

2012 ◽  
Vol 10 (02) ◽  
pp. 207-214 ◽  
Author(s):  
A. F. M. TER ELST ◽  
JOACHIM REHBERG
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
General Setting ◽  
Mixed Boundary Conditions ◽  
Regularity Conditions ◽  
Boundary Part

In this paper, we prove L∞-estimates for solutions of divergence operators in the case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it.

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