Oblique derivative problem for parabolic operators with VMO coefficients
The paper deals with the regular oblique derivative problem \[ \left\{ \begin{aligned} u_t-\sum_{i,j=1}^n a_{ij}(x,t) D_{ij}u=f(x,t)\quad &\text{ a.e. in} Q_T,\\ u(x,0)=\varphi(x) &\quad \text{on} \Omega,\\ \sum_{i=1}^n \ell_i(x,t)D_i u=\psi(x,t)\quad &\text{on} S_T,\end{aligned}\right.\tag \(*\) \] where \(Q_T\) is a cylinder in \({\mathbb R}^n\times {\mathbb R}_+\) of height \(T>0\) with \(C^{1,1}\)-smooth and bounded base \(\Omega\) and lateral surface \(S_T.\) The linear parabolic operator is supposed to be uniformly elliptic and the unit vector field \(\ell(x,t)=(\ell_1(x,t),\ldots,\ell_n(x,t))\) prescribed on \(S_T\) is nowhere tangential to \(S_T.\) The main goal of the author is the study of \((*)\) in the framework of Sobolev spaces \(W^{2,1}_p(Q_T)\) for all values of \(p\in(1,+\infty)\) in the case of discontinuous coefficients \(a_{ij}(x,t).\) More precisely, it is supposed that \(a_{ij}\)'s belong to the Sarason class of functions with vanishing mean oscillation. Precise \(W^{2,1}_p(Q_T)\) a priori estimates are derived for \((*)\) which depend on the VMO-moduli of \(a_{ij}\)'s and generalize the classical ones concerning parabolic operators with continuous coefficients. The approach is based on an explicit representation formula for the derivatives \(D_{ij}u\) and \(L^p\) estimates of singular integral operators with parabolic Calderón-Zygmund kernels and their commutators. As a byproduct, unique strong solvability of \((*)\) in \(W^{2,1}_p(Q_T)\) is proved for all \(p\) in the range \((1,+\infty).\)
explicit representation formula for the derivatives
scientific article; zbMATH DE number 1556765