Morrey regularity of strong solutions to parabolic equations with VMO coefficients

Abstract

We consider a regular oblique derivative problem for a linear parabolic operator $\text{P}$ with VMO principal coefficients. Its unique strong solvability is proved in [15], when $\text{P}\text{u∈}\text{L}{}^{\mathrm{p}}\text{(Q}{}_{\mathrm{T}}\text{)}$. Our goal here is to show that the solution belongs to the parabolic Morrey space W_p,λ^2,1(Q_T), when $\text{P}\text{u∈}\text{L}{}^{\mathrm{p,\lambda }}\text{(Q}{}_{\mathrm{T}}\text{)}$, p∈(1,∞), λ∈(0,n+2), and Q_T is a cylinder in $\text{R}{}_{\mathrm{+}}{}^{\mathrm{n+1}}$. The a priori estimates of the solution are derived through L^p,λ estimates for singular and nonsingular integral operators.