Morrey-type regularity of solutions to parabolic problems with discontinuous data

Abstract

We consider regular oblique derivative problem in cylinder Q _T  = Ω × (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203–220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$, ${p\in(1,\infty)}$. Our aim is to show that the solution belongs to the generalized Sobolev–Morrey space ${W^{2,1}_{p,\omega}(Q_T)}$, when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$, ${p\in (1, \infty)}$, ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$. For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy–Dirichlet problem.