In this paper we consider general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylinders
Ω = {(x0,x,t) ∈ R x Rn-1 x R: x0 > A(x,t)}.
We 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 − ε, 2 + ε) assuming that A(x,·) is uniformly Lipschitz with respect to the time variable and that ||Dt 1/2A||* ≤ ε0 < ∞ for ε0 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(∂Ω) with q−1 + p−1 = 1. As a technical tool, which also is of independent interest, we prove certain square function estimates for solutions to the system.
@article{urn:eudml:doc:41812,
title = {Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains.},
journal = {Collectanea Mathematica},
volume = {57},
year = {2006},
pages = {93-119},
zbl = {1092.35019},
mrnumber = {MR2206182},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41812}
}
Nyström, Kaj. Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains.. Collectanea Mathematica, Tome 57 (2006) pp. 93-119. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41812/