545 research outputs found
Interior error estimate for periodic homogenization
In a previous article about the homogenization of the classical problem of
diff usion in a bounded domain with su ciently smooth boundary we proved that
the error is of order . Now, for an open set with su ciently
smooth boundary and homogeneous Dirichlet or Neuman limits conditions
we show that in any open set strongly included in the error is of order
. If the open set is of polygonal (n=2) or
polyhedral (n=3) boundary we also give the global and interrior error
estimates
Well-posedness and asymptotic behavior of a multidimensional model of morphogen transport
Morphogen transport is a biological process, occurring in the tissue of
living organisms, which is a determining step in cell differentiation. We
present rigorous analysis of a simple model of this process, which is a system
coupling parabolic PDE with ODE. We prove existence and uniqueness of solutions
for both stationary and evolution problems. Moreover we show that the solution
converges exponentially to the equilibrium in topology. We
prove all results for arbitrary dimension of the domain. Our results improve
significantly previously known results for the same model in the case of one
dimensional domain
Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed bvps in exterior domains
This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2013 World Scientific Publishing.Direct segregated systems of boundary-domain integral equations are formulated for the mixed (Dirichlet–Neumann) boundary value problems for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior three-dimensional domain. The boundary-domain integral equation system equivalence to the original boundary value problems and the Fredholm properties and invertibility of the corresponding boundary-domain integral operators are analyzed in weighted Sobolev spaces suitable for infinite domains. This analysis is based on the corresponding properties of the BVPs in weighted Sobolev spaces that are proved as well.The work was supported by the grant EP/H020497/1 \Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients" of the EPSRC, UK
Geometric Aspects of Ambrosetti-Prodi operators with Lipschitz nonlinearities
For Dirichlet boundary conditions on a bounded domain, what happens to the
critical set of the Ambrosetti-Prodi operator if the nonlinearity is only a
Lipschitz map? It turns out that many properties which hold in the smooth case
are preserved, despite of the fact that the operator is not even differentiable
at some points. In particular, a global Lyapunov-Schmidt decomposition of great
convenience for numerical inversion is still available
Stability with respect to domain of the low Mach number limit of compressible viscous fluids
We study the asymptotic limit of solutions to the barotropic Navier-Stokes
system, when the Mach number is proportional to a small parameter \ep \to 0
and the fluid is confined to an exterior spatial domain \Omega_\ep that may
vary with \ep. As , it is shown that the fluid
density becomes constant while the velocity converges to a solenoidal vector
field satisfying the incompressible Navier-Stokes equations on a limit domain.
The velocities approach the limit strongly (a.a.) on any compact set, uniformly
with respect to a certain class of domains. The proof is based on spectral
analysis of the associated wave propagator (Neumann Laplacian) governing the
motion of acoustic waves.Comment: 32 page
A Probabilistic proof of the breakdown of Besov regularity in -shaped domains
{We provide a probabilistic approach in order to investigate the smoothness
of the solution to the Poisson and Dirichlet problems in -shaped domains. In
particular, we obtain (probabilistic) integral representations for the
solution. We also recover Grisvard's classic result on the angle-dependent
breakdown of the regularity of the solution measured in a Besov scale
A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions
Stents are medical devices designed to modify blood flow in aneurysm sacs, in
order to prevent their rupture. Some of them can be considered as a locally
periodic rough boundary. In order to approximate blood flow in arteries and
vessels of the cardio-vascular system containing stents, we use multi-scale
techniques to construct boundary layers and wall laws. Simplifying the flow we
turn to consider a 2-dimensional Poisson problem that conserves essential
features related to the rough boundary. Then, we investigate convergence of
boundary layer approximations and the corresponding wall laws in the case of
Neumann type boundary conditions at the inlet and outlet parts of the domain.
The difficulty comes from the fact that correctors, for the boundary layers
near the rough surface, may introduce error terms on the other portions of the
boundary. In order to correct these spurious oscillations, we introduce a
vertical boundary layer. Trough a careful study of its behavior, we prove
rigorously decay estimates. We then construct complete boundary layers that
respect the macroscopic boundary conditions. We also derive error estimates in
terms of the roughness size epsilon either for the full boundary layer
approximation and for the corresponding averaged wall law.Comment: Dedicated to Professor Giovanni Paolo Galdi 60' Birthda
Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers
The rotated multipliers method is performed in the case of the boundary
stabilization by means of a(linear or non-linear) Neumann feedback. this method
leads to new geometrical cases concerning the "active" part of the boundary
where the feedback is apllied. Due to mixed boundary conditions, these cases
generate singularities. Under a simple geometrical conditon concerning the
orientation of boundary, we obtain a stabilization result in both cases.Comment: 17 pages, 9 figure
Superconductivity in domains with corners
We study the two-dimensional Ginzburg-Landau functional in a domain with
corners for exterior magnetic field strengths near the critical field where the
transition from the superconducting to the normal state occurs. We discuss and
clarify the definition of this field and obtain a complete asymptotic expansion
for it in the large regime. Furthermore, we discuss nucleation of
superconductivity at the boundary
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