Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

Abstract

We continue the development, by reduction to a first-order system for the conormal gradient, of L 2 a priori estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence-form second-order complex elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning a priori almost everywhere nontangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying a posteriori a separate work on bounded domains.

On domain of Poisson operators and factorization for divergence form elliptic operators

“…Note that we have used the condition (2). Recalling that each F k satisfies ∂ t F k + P A 1 F k = G k , we have from the energy calculation and (4.28),…”

“…i,j = j ǫ * a i,j , where j ǫ is a standard mollifier. Note that the conditions (1) (or (4.28)) and (2) are invariant under this mollification. We have already known that H 1 (R d ) = D L 2 (P A (ǫ) ) by [24,Theorem 1.2].…”

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.…”

The method of layer potentials inLpand endpoint spaces for elliptic operators withL∞coefficients

“…The present paper, and its companion article [29], fall into category 1). The paper [28] falls into category 2), and uses our results here to obtain boundedness and solvability results for operators in that category, in which the Carleson measure estimate for the discrepancy is sufficiently small (in this connection, see also the previous work [5], which treats the case p = 2).…”