Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

By P. Auscher, A. Axelsson, S. Hofmann. 2007

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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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0f730f541e9c07d1b16415520792d55c4f4a05c0

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I (2009, methodology): “This fact was proved earlier by Axelsson, Keith and McIntosh [13, Theorem 3.1] elaborating on the technology for the solution of the Kato problem by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian [10]; a more direct proof is proposed in Auscher, Axelsson and McIntosh [8].”
On a quadratic estimate related to the Kato conjecture and boundary value problems (2008, background): “However, we need to exploit the observation made in [2] that one can reduce the action of γt(x) to a subspace, while this is not necessary in [18] or in [4].”
The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ coefficients (2013, background): “4 (allowing complex coefficients) was obtained first in [1], with an alternative proof given in [5].”
Regularity theory for solutions to second order elliptic operators with complex coefficients and the L Dirichlet problem (2016, background): “Finally, there are perturbation results in a variety of special cases, such as [3] and [2]; the first paper shows that solvability in L(2) implies solvability in L for p near 2, and the second paper has L(2)-solvability results for small L∞ perturbations of real elliptic operators when the complex m”
Solvability of elliptic systems with square integrable boundary data (2008, background): “Define the operator N := [ −1 0 0 I ] , which reflects a vector in Rn. Write N+ = 1 2 (1 + N) = [ 0 0 0 I ] for the tangential projection and N− = 1 2 (1− N) = [ 1 0 0 0 ] for the normal projection, so that N = N+ −N−.”
Representation and uniqueness for boundary value elliptic problems via first order systems (2014, methodology): “lowed by N k : Y˙ −1,+ DBe → Y˙ −1 k . The equivalence follows: existence is the same as ontoness of N k and uniqueness is the same as injectivity of N k. The following is an extension of a result in [AAH] for pairs of projections in a non-Hilbert context. Lemma 13.2. Assume that Eis a quasi-Banach ”