The present paper discusses relations between regularity, Dirichlet, and Neumann problems. We investigate the boundary problems for block operators and prove, in particular, that the solvability of the regularity problem does not imply the solvability of the dual Dirichlet problem for general elliptic operators with complex bounded measurable coefficients. This is strikingly different from the case of real operators, for which such an implication was established in 1993 by C. Kenig, J. Pipher [Invent. Math. 113 (3) (1993) 447-509] and since then has served as an integral part of many results.
Bibliographical noteFunding Information:
1 The author was partially supported by the NSF grant DMS 0758500 and the Alfred P. Sloan Foundation.
- Dirichlet problem
- Elliptic equation
- Neumann problem
- Rough coefficients