Resumen de Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains
Kaj Nyström
In this paper we consider general second order, symmetric an and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylinders ! = {(x0, x, t) " R × Rn-1 × R : x0 > A(x, t)}. We pro prove solvability of Dirichlet, Neumann as well as regularity type problems with data in Lp and Lp 1,1/2 (the parabolic Sobolev space having tangential (spatial) gradients and half a time derivative in Lp) for p " (2 - !, 2 + !) assuming that thatA (x, ·) is uniformly Lipschitz with respect to the time variable ariable and that %Dt 1/2A%" & !0 < ' for !0 small enough (%Dt 1/2A%" is the parabolic BMO-norm of a half-derivative in time). We also prove a general structural theorem (duality theorem between Dirichlet and regularity problems) stating that if the Dirichlet problem is solvable in Lp with the relevant bound on the parabolic non-tangential maximal function then the regularity problem can be solved with data in Lq 1,1/2("!) with q-1 + p-1 = 1. As a technical tool, which also is of independ independent interest, we prove certain square function estimates for solutions to the system.
