Links to my papers on MathSciNet and arXiv (A. Axelsson) or ArXiv (A. Rosén)

Link to my PhD thesis in the Australian digital theses database

Erdös number = 4 :
Andreas Rosén - Pascal Auscher - Terence Tao - Vitaly Bergelson - Paul Erdös,
or
Andreas Rosén - Alan McIntosh - Stephen Semmes - Svante Janson - Paul Erdös

34. (with Auscher, P. and Morris, A.) Quadratic estimates for degenerate elliptic systems on manifolds with lower Ricci curvature bounds and boundary value problems. Preprint.
Abstract Weighted quadratic estimates are proved for certain bisectorial first-order differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius. As compared to earlier results, Ricci curvature is only assumed to be bounded from below, and the weight is only assumed to be locally in A2. The Kato square root estimate is proved under this weaker assumption. On compact Lipschitz manifolds we prove solvability estimates for solutions to degenerate elliptic systems with not necessarily self-adjoint coefficients, and with Dirichlet, Neumann and Atiyah--Patodi--Singer boundary conditions.
33. (with Helsing, J. and Karlsson, A.) An efficient full wave solver for eddy currents. To appear in Computers and Mathematics with Applications. https://authors.elsevier.com/sd/article/S0898122122004412
Abstract An integral equation reformulation of the Maxwell transmission problem is presented. The reformulation uses techniques such as tuning of free parameters and augmentation of close-to-rank-deficient operators. It is designed for the eddy current regime and works both for surfaces of genus 0 and 1. Well-conditioned systems and field representations are obtained despite the Maxwell transmission problem being ill-conditioned for genus 1 surfaces due to the presence of Neumann eigenfields. Furthermore, it is shown that these eigenfields, for ordinary conductors in the eddy current regime, are different from the more well-known Neumann eigenfields for superconductors. Numerical examples, based on the reformulation, give an unprecedented 13-digit accuracy both for transmitted and scattered fields.
34. (with Hytönen, T.) Causal sparse domination of Beurling maximal regularity operators. To appear in Journal d'Analyse Mathématique. Full text link.
Abstract We prove boundedness of Calder\'on--Zygmund operators acting in Banach functions spaces on domains, defined by the L_1 Carleson functional and L_q (1<q<\infty) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.
33. (with Helsing, J. and Karlsson, A.) Comparison of integral equations for the Maxwell transmission problem with general permittivities. To appear in Advances in Computational Mathematics.
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 using eight scalar surface densities.
32. (with Helsing, J.) Dirac integral equations for dielectric and plasmonic scattering. Integral Equations and Operator Theory 93 (2021), no. 5, Paper No. 48, DOI 10.1007/s00020-021-02657-1
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. Geometric multivector analysis. From Grassmann to Dirac. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser/Springer, Cham, 2019, xii+465 pp. ISBN: 978-3-030-31410-1; 978-3-030-31411-8
Abstract:  Springer link.
30. Boosting the Maxwell double layer potential using a right spin factor. Integral Equations and Operator Theory 91 (2019), no. 3, Art. 29, 25 pp.
 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.) Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions. Communications in Partial Differential Equations 44 (2019), no. 12, 1253–1284.
Abstract: On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator D B   in L 2   depends Riesz continuously on L ∞   perturbations of local boundary conditions B  . The Lipschitz bound for the map B→D B (1+D 2 B ) −12    depends on Lipschitz smoothness and ellipticity of 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.
28. (with Nursultanov, M.) Evolution of time-harmonic electromagnetic and acoustic waves along waveguides. Integral Equations and Operator Theory 90 (2018), no. 5, Art. 53, 32 pp.
Abstract: We study time-harmonic electromagnetic and acoustic waveguides, modeled by an in nite cylinder with a non-smooth cross section. We introduce an in nitesimal generator for the wave evolution along the cylinder and prove estimates of the functional calculi of these rst order non-self adjoint di erential 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. A spin integral equation for electromagnetic and acoustic scattering. Applicable Analysis 96 (2017), no. 13, 2250-2266.
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.) Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric. Mathematische Annalen 370 (2018), no. 1-2, 863-915.
Abstract: We prove that the Atiyah-Singer Dirac operator D g   in L 2   depends Riesz continuously on L ∞   perturbations of complete metrics g  on a smooth manifold. The Lipschitz bound for the map g→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.
25. (with Hytönen, T.) Bounded variation approximation of L_p dyadic martingales and solutions to elliptic equations. Journal of the European Mathematical Society 20 (2018), no. 8, 1819–1850.
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. A local Tb theorem for matrix weighted paraproducts. Revista Matemática Iberoamericana 32 (2016), no. 4, 1259-1276.
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.) Approximate and exact extensions of Lebesgue boundary functions. Unpublished and contained in publication 25.
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.) Boundary value problems for degenerate elliptic equations. Annales scientifiques de l'ENS 48 (2015), no. 4, 951-1000.
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.

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.

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.