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 $L_2$ for general operators of the form $B_1D_1+D_2B_2$, where $D_i$ are partially elliptic constant coefficient homogeneous first order self-adjoint differential operators with orthogonal ranges, and $B_i$ 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 $B_1$ and $B_2$ are not assumed to be related in any way. We show how these operators appear naturally from exterior differential systems with boundary data in $L_2$. We also prove non-tangential maximal function estimates, where our proof needs only off-diagonal decay of resolvents in $L_2$, unlike earlier proofs which relied on interpolation and $L_p$ 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 $u$ with gradient in weighted $L_2(\R^{1+n}_+,t^{\alpha}dtdx)$ also in the case $|\alpha|<1$. In the end point cases $\alpha= \pm 1$, 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

Approximating Semi-Matchings in Streaming and in Two-Party Communication

We study the communication complexity and streaming complexity of approximating unweighted semi-matchings. A semi-matching in a bipartite graph G = (A, B, E), with n = |A|, is a subset of edges S that matches all A vertices to B vertices with the goal usually being to do this as fairly as possible. While the term 'semi-matching' was coined in 2003 by Harvey et al. [WADS 2003], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 <= \epsilon <= 1 uses space O(n^{1+\epsilon}) and computes an O(n^{(1-\epsilon)/2})-approximation to the semi-matching problem. Furthermore, with O(log n) passes it is possible to compute an O(log n)-approximation with space O(n). In the one-way two-party communication setting, we show that for every \epsilon > 0, deterministic communication protocols for computing an O(n^{1/((1+\epsilon)c + 1)})-approximation require a message of size more than cn bits. We present two deterministic protocols communicating n and 2n edges that compute an O(sqrt(n)) and an O(n^{1/3})-approximation respectively. Finally, we improve on results of Harvey et al. [Journal of Algorithms 2006] and prove new links between semi-matchings and matchings. While it was known that an optimal semi-matching contains a maximum matching, we show that there is a hierarchical decomposition of an optimal semi-matching into maximum matchings. A similar result holds for semi-matchings that do not admit length-two degree-minimizing paths.Comment: This is the long version including all proves of the ICALP 2013 pape

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 $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ 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