The regularity and Neumann problem for non-symmetric elliptic operators

Abstract

Abstract. We establish optimal L p bounds for the non-tangential maximal function of the gradient of the solution to a second-order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the vertical variable, on the domain above a Lipschitz graph in the plane, in terms of the L p -norm at the boundary of the tangential derivative of the Dirichlet data, or of the Neumann data.

Elliptic equations in the plane satisfying a Carleson measure condition

“…Possibly an even more interesting question would be whether or not the Neumann and regularity problems can be shown to hold for some (small) exponent p when c 1 is only assumed to be finite. Given [12] and [11] one would conjecture this is the case, but again our methods are not powerful enough to do this. To prove this conjecture via the methods used here, one would require better knowledge of the constants involved our estimates.…”

“…This would prove that (R) A p holds; that (N) A p holds would then follow by considering a conjugate for a solution to (2.1) defined similarly to (2.5) (see [12]). …”

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

“…It is clear from (7) that U satisfies the Laplace equation in the quadrants t > 0, ±x > 0. Furthermore, calculating ∂ x U (t, 0±) from (3) and ∂ t U (t, 0) from (4) shows that ∂ x U (t, 0+) − ∂ x U (t, 0−) = 2k∂ t U (t, 0) for t > 0. This proves that divA k ∇U = 0 in R 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.…”

“…of non-symmetric coefficient matrices with a jump at x = 0 was studied and shown to provide counterexamples to well-posedness for certain values of the parameter k ∈ R. More precisely, the following theorem was proved in [3, Theorem (3.2.1)] and [4,Appendix]. …”