302 research outputs found
Continuous dependence on the Soret coefficient for double diffusive convection in Darcy flow
AbstractIn this paper, the Darcy model is used to describe the double diffusive flow of a fluid containing a solute. Continuous dependence of the solution on the Soret coefficient is established
A second eigenvalue bound for the Dirichlet Schroedinger operator
Let be the th eigenvalue of the Schr\"odinger
operator with Dirichlet boundary conditions on a bounded domain and with the positive potential . Following the spirit of the
Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the
spherically rearranged potential , we prove that . Here denotes the ball, centered at the
origin, that satisfies the condition .
Further we prove under the same convexity assumptions on a spherically
symmetric potential , that decreases
when the radius of the ball increases.
We conclude with several results about the first two eigenvalues of the
Laplace operator with respect to a measure of Gaussian or inverted Gaussian
density
Two isoperimetric inequalities for the Sobolev constant
In this note we prove two isoperimetric inequalities for the sharp constant
in the Sobolev embedding and its associated extremal function. The first such
inequality is a variation on the classical Schwarz Lemma from complex analysis,
similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini,
and Ransford, while the second generalises an isoperimetric inequality for the
first eigenfunction of the Laplacian due to Payne and Rayner.Comment: 11 page
Continuous Dependence on Initial-Time Geometry for a Thermoelastic System with Sign-Indefinite Elasticities
AbstractContinuous dependence of the solution to the equations of dynamical linear thermoelasticity on the initial-time geometry is investigated. Using a Lagrange identity method and assuming only the major symmetry on the elasticities, but no sign-definiteness, we derive inequalities establishing that appropriately constrained solutions depend Hölder continuously on perturbations of the initial geometry
On the lowest eigenvalue of Laplace operators with mixed boundary conditions
In this paper we consider a Robin-type Laplace operator on bounded domains.
We study the dependence of its lowest eigenvalue on the boundary conditions and
its asymptotic behavior in shrinking and expanding domains. For convex domains
we establish two-sided estimates on the lowest eigenvalues in terms of the
inradius and of the boundary conditions
Logarithmically-concave moment measures I
We discuss a certain Riemannian metric, related to the toric Kahler-Einstein
equation, that is associated in a linearly-invariant manner with a given
log-concave measure in R^n. We use this metric in order to bound the second
derivatives of the solution to the toric Kahler-Einstein equation, and in order
to obtain spectral-gap estimates similar to those of Payne and Weinberger.Comment: 27 page
Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians
We consider the problem of minimising the th eigenvalue, , of
the (-)Laplacian with Robin boundary conditions with respect to all domains
in of given volume . When , we prove that the second
eigenvalue of the -Laplacian is minimised by the domain consisting of the
disjoint union of two balls of equal volume, and that this is the unique domain
with this property. For and , we prove that in many cases a
minimiser cannot be independent of the value of the constant in the
boundary condition, or equivalently of the volume . We obtain similar
results for the Laplacian with generalised Wentzell boundary conditions .Comment: 16 page
A Pinhole Camera for Radiation Research in Dentistry
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/68035/2/10.1177_00220345620410013201.pd
When Does Eddy Viscosity Damp Subfilter Scales Sufficiently?
Large eddy simulation (LES) seeks to predict the dynamics of spatially filtered turbulent flows. The very essence is that the LES-solution contains only scales of size ≥Δ, where Δ denotes some user-chosen length scale. This property enables us to perform a LES when it is not feasible to compute the full, turbulent solution of the Navier-Stokes equations. Therefore, in case the large eddy simulation is based on an eddy viscosity model we determine the eddy viscosity such that any scales of size <Δ are dynamically insignificant. In this paper, we address the following two questions: how much eddy diffusion is needed to (a) balance the production of scales of size smaller than Δ; and (b) damp any disturbances having a scale of size smaller than Δ initially. From this we deduce that the eddy viscosity νe has to depend on the invariants q = ½tr(S^2) and r =−⅓tr(S^3) of the (filtered) strain rate tensor S. The simplest model is then given by νe = 3/2(Δ/π)^2|r|/q. This model is successfully tested for a turbulent channel flow (Reτ = 590).
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