We define various notions of Lagrangian solution in physical space for 3-d incompressible geostrophic system with free upper boudary under different conditions for initial data, then prove their existence via the minimization with respect to a geostrophic functional, generalizing the the work of [5] and [11] to the case of free upper boundary. As a byproduct of our proof, we obtain the existenc...

(1.2) (∂t + V ∂k)ρ+ ρdivV = 0, divV = ∂kV k in D, where V k = δvi = vk and we use the summation convention over repeated upper and lower indices. Here the velocity V = (V , ..., V ), the density ρ and the domain D = ∪0≤t≤T {t}× Dt, Dt ⊂ R are to be determined. The pressure p = p (ρ) is assumed to be a given strictly increasing smooth function of the density. The boundary ∂Dt moves with the velo...

We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing test configurations constructed from local conformal solut...

About 35 samples were collected from lower and upper parts of the proposed Frasnianamennian boundary in three sections of Upper Devonian marine sediments of the Kerman province. The biostratigraphical and geochemical studies indicate that about 45% extinction of brachiopods probably caused by the Frasnian-Famennian bioevents, which is slightly higher than thr rate of Devonian-Carboniferous exti...

We sketch a proof of the fact that the Gromov boundary of a hyperbolic group does not have a global cut point if it is connected. This implies, by a theorem of Bestvina and Mess, that the boundary is locally connected if it is connected. The object of this note is to sketch a proof of the following theorem: Theorem 1. If ? is a one ended hyperbolic group, then the Gromov boundary @? of ? does n...

Let A be a bounded subset of R. We give an upper bound on the volume of the symmetric difference of A and f(A) where f is a translation, a rotation, or the composition of both, a rigid motion. The volume is measured by the d-dimensional Hausdorff measure, which coincides with the Lebesgue measure for Lebesgue measurable sets. We bound the volume of the symmetric difference of A and f(A) in term...

dalichai formation at the rey-abad section (northeast shahroud) along the principal road of the mayamey-jajarm has a thickness of 166 meters .in this section, middle to upper jurassic carbonate rocks are well exposed and is typically composed of massive to medium-bedded limestones with thin shale interbeds. in the studied area and most places of this region, the lower boundary of dalichai forma...

We study symmetry and quantitative approximate for an overdetermined problem involving the fractional torsion in a bounded open set ? ?<!-- ? -...

in this article, we investigate sufficient conditions for existence of maximal and minimalsolutions to a coupled system of highly nonlinear differential equations of fractional order with mixedtype boundary conditions. to achieve this goal, we apply monotone iterative technique togetherwith the method of upper and lower solutions. also an error estimation is given to check theaccuracy of the me...