264 research outputs found
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2011 ElsevierFor functions from the Sobolev space H^s(\Omega), 1/2 < s < 3/2 , definitions of non-unique generalized and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the generalized co-normal derivative inherent non-uniqueness are given. It is shown, that the canonical co-normal derivatives, although de¯ned on a more narrow function class than the generalized ones, are continuous extensions of the classical co-norma derivatives. Some new results about trace operator estimates and Sobolev spaces haracterizations, are also presented
Adaptive FE-BE Coupling for Strongly Nonlinear Transmission Problems with Coulomb Friction
We analyze an adaptive finite element/boundary element procedure for scalar
elastoplastic interface problems involving friction, where a nonlinear
uniformly monotone operator such as the p-Laplacian is coupled to the linear
Laplace equation on the exterior domain. The problem is reduced to a
boundary/domain variational inequality, a discretized saddle point formulation
of which is then solved using the Uzawa algorithm and adaptive mesh refinements
based on a gradient recovery scheme. The Galerkin approximations are shown to
converge to the unique solution of the variational problem in a suitable
product of L^p- and L^2-Sobolev spaces.Comment: 27 pages, 3 figure
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
Quantum graphs with singular two-particle interactions
We construct quantum models of two particles on a compact metric graph with
singular two-particle interactions. The Hamiltonians are self-adjoint
realisations of Laplacians acting on functions defined on pairs of edges in
such a way that the interaction is provided by boundary conditions. In order to
find such Hamiltonians closed and semi-bounded quadratic forms are constructed,
from which the associated self-adjoint operators are extracted. We provide a
general characterisation of such operators and, furthermore, produce certain
classes of examples. We then consider identical particles and project to the
bosonic and fermionic subspaces. Finally, we show that the operators possess
purely discrete spectra and that the eigenvalues are distributed following an
appropriate Weyl asymptotic law
Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients
For functions from the Sobolev space , 1/2<s<3/2, definitions of
non-unique generalised and unique canonical co-normal derivative are
considered, which are related to possible extensions of a partial differential
operator and its right hand side from the domain , where they are
prescribed, to the domain boundary, where they are not. Revision of the
boundary value problem settings, which makes them insensitive to the co-normal
derivative inherent non-uniqueness are given. Some new facts about trace
operator estimates, Sobolev spaces characterisations, and solution regularity
of PDEs with non-smooth coefficients are also presented.Comment: This is the version updated after the content was published in 2
papers, and the two parts of this version correspond to these 2 publication
A note on maximal estimates for stochastic convolutions
In stochastic partial differential equations it is important to have pathwise
regularity properties of stochastic convolutions. In this note we present a new
sufficient condition for the pathwise continuity of stochastic convolutions in
Banach spaces.Comment: Minor correction
Nonexistence of self-similar singularities for the 3D incompressible Euler equations
We prove that there exists no self-similar finite time blowing up solution to
the 3D incompressible Euler equations. By similar method we also show
nonexistence of self-similar blowing up solutions to the divergence-free
transport equation in . This result has direct applications to the
density dependent Euler equations, the Boussinesq system, and the
quasi-geostrophic equations, for which we also show nonexistence of
self-similar blowing up solutions.Comment: This version refines the previous one by relaxing the condition of
compact support for the vorticit
Spectral Duality for Planar Billiards
For a bounded open domain with connected complement in
and piecewise smooth boundary, we consider the Dirichlet Laplacian
on and the S-matrix on the complement . We
show that the on-shell S-matrices have eigenvalues converging to 1
as exactly when has an eigenvalue at energy
. This includes multiplicities, and proves a weak form of
``transparency'' at . We also show that stronger forms of transparency,
such as having an eigenvalue 1 are not expected to hold in
general.Comment: 33 pages, Postscript, A
An alternative approach to regularity for the Navier-Stokes equations in critical spaces
In this paper we present an alternative viewpoint on recent studies of
regularity of solutions to the Navier-Stokes equations in critical spaces. In
particular, we prove that mild solutions which remain bounded in the space
do not become singular in finite time, a result which was proved
in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a
different approach. We use the method of "concentration-compactness" +
"rigidity theorem" which was recently developed by C. Kenig and F. Merle to
treat critical dispersive equations. To the authors' knowledge, this is the
first instance in which this method has been applied to a parabolic equation.
We remark that we have restricted our attention to a special case due only to
a technical restriction, and plan to return to the general case (the
setting) in a future publication.Comment: 41 page
- …