28,074 research outputs found
Causality and the AdS Dirichlet problem
The (planar) AdS Dirichlet problem has previously been shown to exhibit
superluminal hydrodynamic sound modes. This problem is defined by bulk
gravitational dynamics with Dirichlet boundary conditions imposed on a rigid
timelike cut-off surface. We undertake a careful examination of this set-up and
argue that, in most cases, the propagation of information between points on the
Dirichlet hypersurface is nevertheless causal with respect to the induced light
cones. In particular, the high-frequency dynamics is causal in this sense.
There are however two exceptions and both involve boundary gravitons whose
propagation is not constrained by the Einstein equations. These occur in i)
AdS, where the boundary gravitons generally do not respect the induced
light cones on the boundary, and ii) Rindler space, where they are related to
the infinite speed of sound in incompressible fluids. We discuss implications
for the fluid/gravity correspondence with rigid Dirichlet boundaries and for
the black hole membrane paradigm.Comment: 29 pages, 5 figures. v2: added refs. v3: minor clarification
Effective Hamiltonians for Thin Dirichlet Tubes with Varying Cross-Section
We show how to translate recent results on effective Hamiltonians for quantum
systems constrained to a submanifold by a sharply peaked potential to quantum
systems on thin Dirichlet tubes. While the structure of the problem and the
form of the effective Hamiltonian stays the same, the difficulties in the
proofs are different.Comment: 6 pages, 1 figur
On a Bernoulli problem with geometric constraints
A Bernoulli free boundary problem with geometrical constraints is studied.
The domain \Om is constrained to lie in the half space determined by and its boundary to contain a segment of the hyperplane where
non-homogeneous Dirichlet conditions are imposed. We are then looking for the
solution of a partial differential equation satisfying a Dirichlet and a
Neumann boundary condition simultaneously on the free boundary. The existence
and uniqueness of a solution have already been addressed and this paper is
devoted first to the study of geometric and asymptotic properties of the
solution and then to the numerical treatment of the problem using a shape
optimization formulation. The major difficulty and originality of this paper
lies in the treatment of the geometric constraints
A Radial Basis Function Method for Solving PDE Constrained Optimization Problems
In this article, we apply the theory of meshfree methods to the problem of PDE constrained optimization. We derive new collocation-type methods to solve the distributed control problem with Dirichlet boundary conditions and the Neumann boundary control problem, both involving Poisson's equation. We prove results concerning invertibility of the matrix systems we generate, and discuss a modication to guarantee invertibility. We implement these methods using MATLAB, and produce numerical results to demonstrate the methods' capability. We also comment on the methods' effectiveness in comparison to the widely-used finite element formulation of the problem, and make some recommendations as to how this work may be extended
A Bayesian nonparametric approach to log-concave density estimation
The estimation of a log-concave density on is a canonical
problem in the area of shape-constrained nonparametric inference. We present a
Bayesian nonparametric approach to this problem based on an exponentiated
Dirichlet process mixture prior and show that the posterior distribution
converges to the log-concave truth at the (near-) minimax rate in Hellinger
distance. Our proof proceeds by establishing a general contraction result based
on the log-concave maximum likelihood estimator that prevents the need for
further metric entropy calculations. We also present two computationally more
feasible approximations and a more practical empirical Bayes approach, which
are illustrated numerically via simulations.Comment: 39 pages, 17 figures. Simulation studies were significantly expanded
and one more theorem has been adde
Free boundary regularity for a multiphase shape optimization problem
In this paper we prove a regularity result in dimension two
for almost-minimizers of the constrained one-phase Alt-Caffarelli and the
two-phase Alt-Caffarelli-Friedman functionals for an energy with variable
coefficients. As a consequence, we deduce the complete regularity of solutions
of a multiphase shape optimization problem for the first eigenvalue of the
Dirichlet-Laplacian up to the fixed boundary. One of the main ingredient is a
new application of the epiperimetric-inequality of Spolaor-Velichkov [CPAM,
2018] up to the boundary. While the framework that leads to this application is
valid in every dimension, the epiperimetric inequality is known only in
dimension two, thus the restriction on the dimension
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