Oblique derivative problem for parabolic operators with VMO coefficients

Abstract

Let Q T be a cylinder in R n ×R + of height T>0 with C 1,1 smooth base Ω and lateral boundary S T . Unique strong solvability in Sobolev spaces W 2,1 p (Q T ), 1<p<∞, is proved for the regular oblique derivative problem $$$$ in the case of {VMO} coefficients a ij (x,t) of the uniformly parabolic operator. The approach is based on $L^p$ estimates of singular integral operators with parabolic Calderon–Zygmund kernels and their commutators.