TY - JOUR
T1 - The Dirichlet problem in convex bounded domains for operators with L8-coefficients
AU - Hieber, Matthias
AU - Wood, Ian
N1 - Wood, Ian; Hieber, M., (2007) 'The Dirichlet problem in convex bounded domains for operators with L8-coefficients', Differential and Integral Equations 20 pp.721-734
RAE2008
PY - 2007/7/1
Y1 - 2007/7/1
N2 - Consider the Dirichlet problem for elliptic and parabolic equations in non-divergence form with variable coefficients in convex bounded domains of Rn. We prove solvability of the elliptic problem and maximal Lq-Lp-estimates for the solution of the parabolic problem provided the coefficients aij∈L∞ satisfy a Cordes condition and p∈(1,2] is close to 2. This implies that in two dimensions, i.e., n=2, the elliptic Dirichlet problem is always solvable if the associated operator is uniformly strongly elliptic, and p∈(1,2] is close to 2, for maximal Lq-Lp-regularity in the parabolic case an additional assumption on the growth of the coefficients is needed.
AB - Consider the Dirichlet problem for elliptic and parabolic equations in non-divergence form with variable coefficients in convex bounded domains of Rn. We prove solvability of the elliptic problem and maximal Lq-Lp-estimates for the solution of the parabolic problem provided the coefficients aij∈L∞ satisfy a Cordes condition and p∈(1,2] is close to 2. This implies that in two dimensions, i.e., n=2, the elliptic Dirichlet problem is always solvable if the associated operator is uniformly strongly elliptic, and p∈(1,2] is close to 2, for maximal Lq-Lp-regularity in the parabolic case an additional assumption on the growth of the coefficients is needed.
M3 - Article
SN - 0893-4983
VL - 20
SP - 721
EP - 734
JO - Differential and Integral Equations
JF - Differential and Integral Equations
IS - 7
ER -