Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains.

Nyström, Kaj. "Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains.." Collectanea Mathematica 57.1 (2006): 93-119. <http://eudml.org/doc/41812>.

@article{Nyström2006,
abstract = {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 cylindersΩ = \{(x0,x,t) ∈ R x Rn-1 x R: x0 &gt; A(x,t)\}.We prove solvability of Dirichlet, Neumann as well as regularity type problems with data in Lp and Lp1,1/2 (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 ||Dt1/2A||* ≤ ε0 &lt; ∞ for ε0 small enough (||Dt1/2A||* 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 Lq1,1/2(∂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.},
author = {Nyström, Kaj},
journal = {Collectanea Mathematica},
keywords = {Ecuaciones parabólicas; Problemas de valor de frontera; Problema de Dirichlet; Espacios LP; Regularidad; Medidas de Carleson; Integrales singulares; parabolic Sobolev space; non-tangential maximal function; certain square function; Carleson measure},
language = {eng},
number = {1},
pages = {93-119},
title = {Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains.},
url = {http://eudml.org/doc/41812},
volume = {57},
year = {2006},
}

TY - JOUR
AU - Nyström, Kaj
TI - Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains.
JO - Collectanea Mathematica
PY - 2006
VL - 57
IS - 1
SP - 93
EP - 119
AB - 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 cylindersΩ = {(x0,x,t) ∈ R x Rn-1 x R: x0 &gt; A(x,t)}.We prove solvability of Dirichlet, Neumann as well as regularity type problems with data in Lp and Lp1,1/2 (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 ||Dt1/2A||* ≤ ε0 &lt; ∞ for ε0 small enough (||Dt1/2A||* 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 Lq1,1/2(∂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.
LA - eng
KW - Ecuaciones parabólicas; Problemas de valor de frontera; Problema de Dirichlet; Espacios LP; Regularidad; Medidas de Carleson; Integrales singulares; parabolic Sobolev space; non-tangential maximal function; certain square function; Carleson measure
UR - http://eudml.org/doc/41812
ER -