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

Abstract

We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L 2 for small complex L ∞ 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.

Boundedness and applications of singular integrals and square functions: a survey

“…These include L 2 bounds for layer potentials associated to complex divergence form elliptic operators [4,55], and the development of an L 2 functional calculus of certain perturbed Dirac operators and other first order elliptic systems [5][6][7]14]. The layer potential bounds, and the existence of a bounded holomorphic functional calculus for first order elliptic systems, were each then applied to obtain L 2 solvability results for elliptic boundary value problems.…”

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

“…In this section we explicitly calculate the basic operators we need in order to solve the BVP's with the boundary equation method from [2]. As in [2, equation (1.5)], we rewrite equation (1) for U as the equivalent first order system divA k F = 0 and curlF = 0 for the vector field F = ∇U .…”

“…In this section we use the Cauchy integrals from Theorem 2.1 to solve BVP's, following the boundary equation method described in [2]. …”

“…In the half plane, for real but non-symmetric coefficients, L p solvability of the Dirichlet problem for sufficiently large p was obtained by Kenig, Koch, Pipher and Toro [3], and L p solvability of the Neumann and regularity problems, for sufficiently small p, was proved by Kenig and Rule [4]. In R n , two boundary equation methods have been studied by Alfonseca, Auscher, Axelsson, Hofmann and Kim [1] and by Auscher, Axelsson and Hofmann [2] where, among other things, these BVP's were proved to be well posed in L 2 for small complex L ∞ perturbations of real symmetric coefficients.…”

“…In this paper, we shall use the boundary equation method from Auscher, Axelsson and Hofmann [2]. The Cauchy integral method used here for the Neumann and regularity problems actually uses a vector valued kernel K(t, x; y), and constructs the gradient vector field F = ∇U rather than U itself.…”