Link to my PhD
thesis in the Australian digital theses database

33. (with Helsing, J. and Karlsson, A.)

Abstract Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using32. (with Helsing, J.)eight scalar surface densities.

Abstract: A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for general complex-valued permittivities, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, the new formulation applies to scattering also in three dimensions, where the theory suggests that it will perform equally well from a numerical point of view.31.

Abstract: Springer link.30.

Abstract: 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.29. (with Bandara, L.)

Abstract: On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator28. (with Nursultanov, M.)D B inL 2 depends Riesz continuously onL ∞ perturbations of local boundary conditionsB . The Lipschitz bound for the mapB→D B (1+D 2 B ) −12 depends on Lipschitz smoothness and ellipticity ofB 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.

Abstract: We study time-harmonic electromagnetic and acoustic waveguides, modeled by an innite cylinder with a non-smooth cross section. We introduce an innitesimal generator for the wave evolution along the cylinder and prove estimates of the functional calculi of these rst order non-self adjoint dierential operators with non-smooth coecients. Applying our new functional calculus, we obtain a one-to-one correspondence between polynomially bounded time-harmonic waves and functions in appropriate spectral subspaces.27.

Abstract: We present a new integral equation for solving the Maxwell scattering problem against a perfect conductor. The very same algorithm also applies to sound-soft as well as sound-hard Helmholtz scattering, and in fact the latter two can be solved in parallel in three dimensions. Our integral equation does not break down at interior spurious resonances, and uses spaces of functions without any algebraic or differential constraints. The operator to invert at the boundary involves a singular integral operator closely related to the three dimensional Cauchy singular integral, and is bounded on natural function spaces and depend analytically on the wave number. Our operators act on functions with pairs of complex two by two matrices as values, using a spin representation of the fields.26. (with Bandara, L. and McIntosh, A.)

Abstract: We prove that the Atiyah-Singer Dirac operator25. (with Hytönen, T.)D g inL 2 depends Riesz continuously onL ∞ perturbations of complete metricsg on a smooth manifold. The Lipschitz bound for the mapg→D g (1+D 2 g ) −12 depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

Abstract: We prove continuity and surjectivity of the trace map onto L_p(R^n), from a space of functions of locally bounded variation, defined by the Carleson functional. The extension map is constructed through a stopping time argument. This extends earlier work by Varopoulos in the BMO case, related to the Corona theorem. We also prove L_p Carleson approximability results for solutions to elliptic non-smooth divergence form equations, which generalize results in the case p=infty by Hofmann, Kenig, Mayboroda and Pipher.24.

Abstract: We prove a local Tb theorem for paraproducts acting on vector valued functions, with matrix weighted averaging operators. The condition on the weight is that its square is in the L_2 associated matrix A_\infty class. We also introduce and use a new matrix reverse Hölder class. This result generalizes the previously known case of scalar weights from the proof of the Kato square root problem, as well as the case of diagonal weights, recently used in the study of boundary value problems for degenerate elliptic equations.23. (with Hytönen, T.)

Abstract: We prove continuity and surjectivity of the trace map onto L_p(R^n), from a space of functions of locally bounded variation, defined by the Carleson functional. The extension map is constructed through a stopping time argument. This extends earlier work by Varopoulos in the $\bmo$ case, related to the Corona theorem.22. (with Auscher, P. and Rule, D.)

Abstract: We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an A2 weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.21. (with Nyström, K.) Cauchy integrals for the p-Laplace equation. Annales Academiae Scientiarum Fennicae Mathematica 39 (2014), 545-565.

Abstract: 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.20. Square function and maximal function estimates for operators beyond divergence form equations. Journal of Evolution Equations 13 (2013), 651-674.

Abstract: 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.19. Cauchy non-integral formulas. Contemporary Mathematics 612 (2014), 163-178. (Proceedings of the El Escorial conference 2012).

Abstract: 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. Rosén. 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. Hytönen and A. Rosén to establish such Cauchy formulas.18. Layer potentials beyond singular integral operators. Publicacions Matemŕtiques 57 (2013), no. 2, 429-454.

Abstract: We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier results, our proof shows that the bounds for the layer potentials are independent of well posedness for the Dirichlet problem and of De Giorgi-Nash local estimates. The layer potential operators are shown to depend holomorphically on the coefficient matrix A\in L_\infty, showing uniqueness of the extension of the operators beyond singular integrals. More precisely, we use functional calculus of differential operators with non-smooth coefficients to represent the layer potential operators as bounded Hilbert space operators. In the presence of Moser local bounds, in particular for real scalar equations and systems that are small perturbations of real scalar equations, these operators are shown to be the usual singular integrals. Our proof gives a new construction of fundamental solutions to divergence form systems, valid also in dimension 2.17. (with Hytönen, T.) On the Carleson duality. Arkiv för Matematik 51 (2013), 293–313.

Abstract: 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.16. (with Auscher, P.) Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II. Analysis & PDE 5-5 (2012), 983--1061.

Abstract: In this article, we continue the
development of new solvability methods for boundary value problems
of Dirichlet, regularity, Neumann type with square integrable data
for divergence form second order, real and complex, elliptic
systems. We work on the unit ball and more generally its
bi-Lipschitz images, and we assume a Carleson condition as
introduced by Dahlberg measuring the discrepancy of the
coefficients to their boundary trace near the boundary. The method
is a reduction to a first order system for the conormal gradient,
use of maximal regularity estimates to represent solutions in
various classes, and Fredholm theory. As compared to our previous
work, we also prove almost everywhere non-tangential convergence
at the boundary for solutions. Furthermore we make a comparison of
our method with the one using non-tangential maximal control of
solutions in the case of real equations. This leads to new results
for real equations such as the well-posedness of the regularity
problem with continuous coefficients satisfying a transversal
square Dini condition.

15. (with Auscher, P.) Remarks on maximal regularity. Progress in Nonlinear Differential Equations and Their Applications 60 (2011), 45–55.

Abstract: We prove weighted estimates for the
maximal regularity operator. Such estimates were motivated by
boundary value problems. We take this opportunity to study a class
of weak solutions to the abstract Cauchy problem. We also
give a new proof of maximal regularity for closed and
maximal accretive operators following from Kato's inequality for
fractional powers and almost orthogonality arguments.

14. (with Auscher, P.) Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I. Inventiones Mathematicae 184 (2011), no. 1, 47-115.

Abstract: We develop new solvability methods
for divergence form second order, real and complex, elliptic
systems above Lipschitz graphs, with $L_2$ boundary data. Our
methods yield full characterization of weak solutions, whose
gradients have $L_2$ estimates of a non-tangential maximal
function or of the square function, via an integral representation
acting on the conormal gradient, with a singular operator-valued
kernel.

The coefficients $A$ may depend on all variables, but are assumed to be close to coefficients $A_0$ that are independent of the coordinate transversal to the boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour under finiteness of $\|A-A_0\|_C$. For example, the non-tangential maximal function of a weak solution is controlled in $L_2$ by the square function of its gradient. This estimate is new for systems in such generality, even for real non-symmetric equations in dimension 3 or higher. The existence of a proof a priori to well-posedness, is also a new fact. As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of $\|A-A_0\|_C$ and well-posedness for $A_0$, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients $A_0$ by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients $A$ is an operational calculus to prove weighted maximal regularity estimates.

The coefficients $A$ may depend on all variables, but are assumed to be close to coefficients $A_0$ that are independent of the coordinate transversal to the boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour under finiteness of $\|A-A_0\|_C$. For example, the non-tangential maximal function of a weak solution is controlled in $L_2$ by the square function of its gradient. This estimate is new for systems in such generality, even for real non-symmetric equations in dimension 3 or higher. The existence of a proof a priori to well-posedness, is also a new fact. As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of $\|A-A_0\|_C$ and well-posedness for $A_0$, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients $A_0$ by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients $A$ is an operational calculus to prove weighted maximal regularity estimates.

13. (with Alfonseca, A.,Auscher, P., Hofmann, S. and Seick, K.) Analyticity of layer potentials and L˛ solvability of boundary value problems for divergence form elliptic equations with complex L^infinity coefficients. Advances in Mathematics 226 (2011), no. 5, 4533-4606.

Abstract: We consider divergence form
elliptic operators of the form $L=-\dv A(x)\nabla$, defined in
$R^{n+1} = \{(x,t)\in R^n \times R \}$, $n \geq 2$, where the
$L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$,
uniformly elliptic, complex and $t$-independent. We show that for
such operators, boundedness and invertibility of the corresponding
layer potential operators on
$L^2(\mathbb{R}^{n})=L^2(\partial\mathbb{R}_{+}^{n+1})$, is stable
under complex, $L^{\infty}$ perturbations of the coefficient
matrix. Using a variant of the $Tb$ Theorem, we also prove that
the layer potentials are bounded and invertible on
$L^2(\mathbb{R}^n)$ whenever $A(x)$ is real and symmetric (and
thus, by our stability result, also when $A$ is complex, $\Vert
A-A^0\Vert_{\infty}$ is small enough and $A^0$ is real, symmetric,
$L^{\infty}$ and elliptic). In particular, we establish
solvability of the Dirichlet and Neumann (and Regularity)
problems, with $L^2$ (resp. $\dot{L}^2_1)$ data, for small complex
perturbations of a real symmetric matrix. Previously, $L^2$
solvability results for complex (or even real but non-symmetric)
coefficients were known to hold only for perturbations of constant
matrices (and then only for the Dirichlet problem), or in the
special case that the coefficients $A_{j,n+1}=0=A_{n+1,j}$, $1\leq
j\leq n$, which corresponds to the Kato square root problem.

12. (with Kou, K.I. and Qian, T.) Hilbert transforms and the Cauchy integral in euclidean spaces. Studia Mathematica 193 (2009), no. 2, 161-187.

Abstract: We generalize the notions of
harmonic conjugate functions and Hilbert transforms to higher
dimensional euclidean spaces, in the setting of differential forms
and the Hodge-Dirac system. These harmonic conjugates are in
general far from being unique, but under suitable boundary
conditions we prove existence and uniqueness of conjugates. The
proof also yields invertibility results for a new class of
generalized double layer potential operators on Lipschitz surfaces
and boundedness of related Hilbert transforms.

11. (with Auscher, P. and McIntosh, A.) On a quadratic estimate related to the Kato conjecture and boundary value problems. Contemporary Mathematics 205 (2010), 105-129. (Proceedings of the El Escorial conference 2008).

Abstract: We provide a direct proof of a
quadratic estimate that plays a central role in the determination
of domains of square roots of elliptic operators and, as
shown more recently, in some boundary value problems with
$L^2$ boundary data. We develop the application to the Kato
conjecture and to a Neumann problem. This quadratic estimate
enjoys some equivalent forms in various settings. This gives
new results in the functional calculus of Dirac type operators on
forms.

10. (with Auscher, P. and McIntosh, A.) Solvability of elliptic systems with square integrable boundary data. Arkiv för Matematik 48 (2010), 253-287.

Abstract: We consider second order elliptic
divergence form systems with complex measurable coefficients $A$
that are independent of the transversal coordinate, and prove that
the set of $A$ for which the boundary value problem with $L_2$
Dirichlet or Neumann data is well posed, is an open set.
Furthermore we prove that these boundary value problems are well
posed when $A$ is either Hermitean, block or constant. Our methods
apply to more general systems of PDEs and as an example we prove
perturbation results for boundary value problems for differential
forms.

9. (with Auscher, P. and Hofmann, S.) Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems. Journal of functional analysis 255 (2008), no. 2, 374-448.

Abstract: We prove that the Neumann,
Dirichlet and regularity problems for divergence form elliptic
equations in the half space are well posed in $L_2$ for small
complex $L_\infty$ perturbations of a coefficient matrix which is
either real symmetric, of block form or constant. All matrices are
assumed to be independent of the transversal coordinate. We solve
the Neumann, Dirichlet and regularity problems through a new
boundary operator method which makes use of operators in the
functional calculus of an underlaying first order Dirac type
operator. We establish quadratic estimates for this Dirac
operator, which implies that the associated Hardy projection
operators are bounded and depend continuously on the coefficient
matrix. We also prove that certain transmission problems for
$k$-forms are well posed for small perturbations of block
matrices.

8. Non unique solutions to boundary value problems for non symmetric divergence form equations. Transactions of the American Mathematical Society 362 (2010), no. 2, 661-672.

Abstract: We calculate explicitly solutions
to the Dirichlet and Neumann boundary value problems in the upper
half plane, for a family of divergence form equations with non
symmetric coefficients with a jump discontinuity. It is shown that
the boundary equation method and the Lax-Milgram method for
constructing solutions may give two different solutions when the
coefficients are sufficiently non symmetric.

7. Transmission problems for Maxwell's equations with weakly Lipschitz interfaces. Mathematical Methods in the Applied Sciences 29 (2006), no. 6, 665-714. (This is an extended version of chapter 5 of my PhD thesis.)

Abstract: We prove sufficient conditions on
material constants, frequency and Lipschitz regularity of
interface for well posedness of a generalized Maxwell transmission
problem in finite energy norms. This is done by embedding
Maxwell's equations in an elliptic Dirac equation, by constructing
the natural trace space for the transmission problem and using
Hodge decompositions for operators $d$ and $\del$ on weakly
Lipschitz domains to prove stability. We also obtain results for
boundary value problems and transmission problems for the
Hodge-Dirac equation and prove spectral estimates for boundary
singular integral operators related to double layer potentials.

6. (with Keith, S. and McIntosh, A.) The Kato square root problem for mixed boundary value problems. Journal of the London Mathematical Society (2) 74 (2006), 113-130.

Abstract: We solve the Kato square root
problem for second order elliptic systems in divergence form under
mixed boundary conditions on Lipschitz domains. This answers a
question posed by J.-L. Lions in 1962. To do this we develop a
general theory of quadratic estimates and functional calculi for
complex perturbations of Dirac-type operators on Lipschitz
domains.

5. (with Keith, S. and McIntosh, A.) Quadratic estimates and functional calculi of perturbed Dirac operators. Inventiones Mathematicae 163 (2006), no. 3, 455-497.

Abstract: We prove quadratic estimates for
complex perturbations of Dirac-type operators, and thereby show
that such operators have a bounded functional calculus. As an
application we show that spectral projections of the Hodge-Dirac
operator on compact manifolds depend analytically on
$L_\infty$ changes in the metric. We also recover a unified proof
of many results in the Calderón program, including the Kato square
root problem and the boundedness of the Cauchy operator on
Lipschitz curves and surfaces.

4. (with McIntosh, A.) Hodge decompositions on weakly Lipschitz domains. In: T. Qian, T. Hempfling, A. \Mc Intosh, F. Sommen (eds.), Advances in Analysis and Geometry, New Developments Using Clifford Algebras, ISBN 3-7643-6661-3, Series Trends in Mathematics, Birkhauser Basel, 2004. (This is an extended version of chapter 4 of my PhD thesis.)

Abstract: We survey the $L_2$ theory of
boundary value problems for exterior and interior derivative
operators $d_{k_1}=d+k_1 e_0\wedg$ and $\del_{k_2}= \del+k_2
e_0\lctr$ on a bounded, weakly Lipschitz domain
$\Omega\subset\R^n$, for $k_1$, $k_2\in\C$. The boundary
conditions are that the field be either normal or tangential at
the boundary. The well-posedness of these problems is related to a
Hodge decomposition of the space $L_2(\Omega)$ corresponding to
the operators $d$ and $\del$. In developing this relationship, we
derive a theory of nilpotent operators in Hilbert space.

3. Oblique and normal transmission problems for Dirac operators with strongly Lipschitz interfaces. Communications in Partial Differential Equations 28 (2003), no. 11-12, 1911-1941. (This is chapter 3 of my PhD thesis.)

Abstract: We investigate transmission
problems with strongly Lipschitz interfaces for the Dirac equation
by establishing spectral estimates on an associated boundary
singular integral operator, the rotation operator. Using Rellich
estimates we obtain angular spectral estimates on both the
essential and full spectrum for general bi-oblique transmission
problems. Specializing to the normal transmission problem, we
investigate transmission problems for Maxwell's equations using a
nilpotent exterior/interior derivative operator. The fundamental
commutation properties for this operator with the two basic
reflection operators are proved. We show how the $L_2$ spectral
estimates are inherited for the domain of the exterior/interior
derivative operator and prove some complementary eigenvalue
estimates. Finally we use a general algebraic theorem to prove a
regularity property

needed for Maxwell's equations.

needed for Maxwell's equations.

2. Transmission problems and boundary operator algebras. Integral Equations and Operator Theory 50 (2004), no. 2, 147-164. (This is chapter 2 of my PhD thesis.)

Abstract: We examine the operator algebra
$\mA$ behind the boundary integral equation method for solving
transmission problems. A new type of boundary integral operator,
the rotation operator, is introduced, which is more appropriate
than operators of double layer type for solving transmission
problems for first order elliptic partial differential equations.
We give a general invertibility criteria for operators in $\mA$ by
defining a Clifford algebra valued Gelfand transform on $\mA$. The
general theory is applied to transmission problems with strongly
Lipschitz interfaces for the two classical elliptic operators
$\overline\partial$ and $\Delta$. We here use Rellich techniques
in a new way to estimate the full complex spectrum of the boundary
integral operators. For $\overline\partial$ we use the associated
rotation operator to solve the Hilbert boundary value problem and
a Riemann type transmission problem. For the Helmholtz equation,
we demonstrate how Rellich estimates give an angular spectral
estimate on the rotation operator, which with the general spectral
mapping properties in $\mA$ translates to a hyperbolic spectral
estimate for the double layer potential operator.

1. (with Grognard, R., Hogan, J. and McIntosh, A.) Harmonic analysis of Dirac operators on Lipschitz domains. Clifford analysis and its applications (Prague, 2000), 231-246, NATO Sci. Ser. II Math. Phys. Chem., 25, Kluwer Acad. Publ., Dordrecht, 2001.

Abstract: We survey some results concerning
Clifford analysis and the $L^2$ theory of boundary value problems
on domains with Lipschitz boundaries. Some novelty is introduced
when using Rellich inequalities to invert boundary operators.