109 research outputs found
Boosting the Maxwell double layer potential using a right spin factor
We construct new spin singular integral equations for solving scattering
problems for Maxwell's equations, both against perfect conductors and in media
with piecewise constant permittivity, permeability and conductivity, improving
and extending earlier formulations by the author. These differ in a fundamental
way from classical integral equations, which use double layer potential
operators, and have the advantage of having a better condition number, in
particular in Fredholm sense and on Lipschitz regular interfaces, and do not
suffer from spurious resonances. The construction of the integral equations
builds on the observation that the double layer potential factorises into a
boundary value problem and an ansatz. We modify the ansatz, inspired by a
non-selfadjoint local elliptic boundary condition for Dirac equations
Square function and maximal function estimates for operators beyond divergence form equations
We prove square function estimates in for general operators of the form
, where are partially elliptic constant coefficient
homogeneous first order self-adjoint differential operators with orthogonal
ranges, and are bounded accretive multiplication operators, extending
earlier estimates from the Kato square root problem to a wider class of
operators. The main novelty is that and are not assumed to be
related in any way. We show how these operators appear naturally from exterior
differential systems with boundary data in . We also prove non-tangential
maximal function estimates, where our proof needs only off-diagonal decay of
resolvents in , unlike earlier proofs which relied on interpolation and
estimates
Cauchy non-integral formulas
We study certain generalized Cauchy integral formulas for gradients of
solutions to second order divergence form elliptic systems, which appeared in
recent work by P. Auscher and A. Ros\'en. These are constructed through
functional calculus and are in general beyond the scope of singular integrals.
More precisely, we establish such Cauchy formulas for solutions with
gradient in weighted also in the case
. In the end point cases , we show how to apply
Carleson duality results by T. Hyt\"onen and A. Ros\'en to establish such
Cauchy formulas
Evolution of time-harmonic electromagnetic and acoustic waves along waveguides
We study time-harmonic electromagnetic and acoustic waveguides, modeled by an
infinite cylinder with a non-smooth cross section. We introduce an
infinitesimal generator for the wave evolution along the cylinder, and prove
estimates of the functional calculi of these first order non-self adjoint
differential operators with non-smooth coefficients. Applying our new
functional calculus, we obtain a one-to-one correspondence between polynomially
bounded time-harmonic waves and functions in appropriate spectral subspaces
On the Carleson duality
As a tool for solving the Neumann problem for divergence form equations,
Kenig and Pipher introduced the space X of functions on the half space, such
that the non-tangential maximal function of their L_2-Whitney averages belongs
to L_2 on the boundary. In this paper, answering questions which arose from
recent studies of boundary value problems by Auscher and the second author, we
find the pre-dual of X, and characterize the pointwise multipliers from X to
L_2 on the half space as the well-known Carleson-type space of functions
introduced by Dahlberg. We also extend these results to L_p generalizations of
the space X. Our results elaborate on the well-known duality between Carleson
measures and non-tangential maximal functions.Comment: The second author has recently changed surname from previous name
Andreas Axelsso
Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions
On a smooth complete Riemannian spin manifold with smooth compact boundary,
we demonstrate that the Atiyah-Singer Dirac operator
in depends Riesz continuously on
perturbations of local boundary conditions . The Lipschitz bound
for the map depends on Lipschitz smoothness
and ellipticity of and bounds on Ricci curvature and its first
derivatives as well as a lower bound on injectivity radius. More generally, we
prove perturbation estimates for functional calculi of elliptic operators on
manifolds with local boundary conditions.Comment: Final versio
Cauchy integrals for the p-Laplace equation in planar Lipschitz domains
We construct solutions to p-Laplace type equations in unbounded Lipschitz
domains in the plane with prescribed boundary data in appropriate fractional
Sobolev spaces. Our approach builds on a Cauchy integral representation formula
for solutions.Comment: Error in proof correcte
Approximate dynamic fault tree calculations for modelling water supply risks
Traditional fault tree analysis is not always sufficient when analysing complex systems. To overcome
the limitations dynamic fault tree (DFT) analysis is suggested in the literature as well as different
approaches for how to solve DFTs. For added value in fault tree analysis, approximate DFT calculations
based on a Markovian approach are presented and evaluated here. The approximate DFT calculations
are performed using standard Monte Carlo simulations and do not require simulations of the full
Markov models, which simplifies model building and in particular calculations. It is shown how to
extend the calculations of the traditional OR- and AND-gates, so that information is available on the
failure probability, the failure rate and the mean downtime at all levels in the fault tree. Two additional
logic gates are presented that make it possible to model a system’s ability to compensate for failures.
This work was initiated to enable correct analyses of water supply risks. Drinking water systems are
typically complex with an inherent ability to compensate for failures that is not easily modelled using
traditional logic gates. The approximate DFT calculations are compared to results from simulations of theorresponding Markov models for three water supply examples. For the traditional OR- and AND-gates, and one gate modelling compensation, the errors in the results are small. For the other gate modelling compensation, the error increases with the number of compensating components. The errors are, however, in most cases acceptable with respect to uncertainties in input data. The approximate DFT calculations improve the capabilities of fault tree analysis of drinking water systems since they provide additional and important information and are simple and practically applicable
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