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

Collectanea Mathematica (2006)

- Volume: 57, Issue: 1, page 93-119
- ISSN: 0010-0757

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topNyström, 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 > 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 < ∞ 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 > 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 < ∞ 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 -