10 research outputs found
Existence theorems for a crystal surface model involving the p-Laplace operator
The manufacturing of crystal films lies at the heart of modern
nanotechnology. How to accurately predict the motion of a crystal surface is of
fundamental importance. Many continuum models have been developed for this
purpose, including a number of PDE models, which are often obtained as the
continuum limit of a family of kinetic Monte Carlo models of crystal surface
relaxation that includes both the solid-on-solid and discrete Gaussian models.
In this paper we offer an analytical perspective into some of these models. To
be specific, we study the existence of a weak solution to the boundary value
problem for the equation - \Delta e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla
u\right)}+au=f, where are given numbers and is a given
function. This problem is derived from a crystal surface model proposed by
J.L.~Marzuola and J.~Weare (2013 Physical Review, E 88, 032403). The
mathematical challenge is due to the fact that the principal term in our
equation is an exponential function of a p-Laplacian. Existence of a
suitably-defined weak solution is established under the assumptions that
, and . Our investigations reveal that the
key to our existence assertion is how to control the set where
-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right) is
Forest plot of OC use and meningioma risk.
<p>*The risk estimates are computed from raw data or abstracted from original studies.</p
Forest plot of HRT use and meningioma risk.
<p>*The risk estimates are computed from raw data or abstracted from original studies.</p
Flow diagram of literature search and selection.
<p>Flow diagram of literature search and selection.</p
Forest plot of menopausal status and meningioma risk.
<p>*The risk estimates are computed from raw data or abstracted from original studies.</p
Forest plot of parity and meningioma risk.
<p>Forest plot of parity and meningioma risk.</p
Forest plot of age at first birth and meningioma risk.
<p>Forest plot of age at first birth and meningioma risk.</p
Characteristic of the included studies in this meta-analysis.
<p>PCC, population-based case-control study; HCC, hospital-based case-control study; NCC, nested case-control study; BIM, body mass index.</p><p><sup>a</sup> Studies were conducted in: (1) USA, (2) Sweden, (3) Denmark, (4) Finland, (5) Norway, (6) the United Kingdom, (7) Australia, (8) France, (9) Canada, (10)Spain, (11)Italy, (12)Germany, (13)Greece, and (14)the Netherlands.</p
Forest plot of age at menopause and meningioma risk.
<p>Forest plot of age at menopause and meningioma risk.</p
Forest plot of age at menarche and meningioma risk.
<p>*The risk estimates are computed from raw data or abstracted from original studies.</p