Solvability of elliptic systems with square integrable boundary data

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 partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.

Evolution of Time-Harmonic Electromagnetic and Acoustic Waves Along Waveguides

“…Even in the static case D 0 = 0, T is only a bi-sectoral operator (see [3]), and L 2 (Ω) bounds of exp(−tT ) and more general functions f (T ) of T , are non-trivial matters. However, in the general non-smooth case, this is well understood from the works of Axelsson et al [5] and Auscher et al [4]. In the present paper we extend these results to the case D 0 = 0 which occurs in general time-harmonic, but non-static, wave propagation in waveguides.…”

“…Then the equation becomes 6) together with the constraint that f ∈ R(D) for each fixed t ∈ R. Since A is a pointwise strictly accretive operator, B is a strictly accretive multiplication operator just like A, see [4,Proposition 3.2]. By the above arguments, equation (2.2) for u implies that f , defined above, solves (2.6).…”

The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

“…We also observe that more recently, in the case 1 although the technology of the Kato problem continues to play a crucial role in the present paper and in [HKMP]. p = 2, the fact that (D 2 ) (with square function estimates) ⇐⇒ (R) 2 (again, up to adjoints), was established explicitly in [AR] (when the domain is the ball, but the proof there carries over to the half-space mutatis mutandi), and is at least implicit in the combination of results in [AAMc,Section 4] and [AAAHK,Estimate (5.3)], and also in [AA,Section 9]. Our main result, Theorem 1.11, generalizes all implications above, at least as far as the t-independent matrices are concerned, under the assumption of De Giorgi-Nash-Moser bounds for solutions.…”

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.…”