Andreas Axelsson

Quadratic estimates and functional calculi of perturbed Dirac operators

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∞ 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.

The Kato square root problem for mixed boundary value problems

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 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.

Harmonic Analysis of Dirac Operators on Lipschitz Domains

We survey some results concerning Clifford analysis and the L2 theory of boundary value problems on domains with Lipschitz boundaries. Some novelty is introduced when using Rellich inequalities to invert boundary operators.

Non unique solutions to boundary value problems for non symmetric divergence form equations

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 ...

Hodge Decompositions on Weakly Lipschitz Domains

We survey the L 2 theory of boundary value problems for exterior and interior derivative operators \( {d_{k1}} = d + {k_1}eo \wedge \) and \( {\delta _{k2}} = \delta + {k_2}eo \) on a bounded, weakly Lipschitz domain \(\Omega \subset {{R}^{n}} \), for k 1, k 2 ∈ 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(Ω) corresponding to the operators d and δ In developing this relationship, we derive a theory of nilpotent operators in Hilbert space.

Oblique and normal transmission problems for Dirac operators with strongly Lipschitz interfaces

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 Maxwells equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L 2spectral 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 Maxwells equations.