Interior W 1,p Regularity and Hölder Continuity of Weak Solutions to a Class of Divergence Kolmogorov Equations with Discontinuous Coefficients

Abstract

We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain \({\Omega \subset \mathbb{R}^{N+1}}\), where the coefficients matrix (aij (z)) is symmetric uniformly positive definite on \({\mathbb{R}^{p_0} (1 \leq p_0 < N)}\). We obtain interior W1,p (1 < p < ∞) regularity and Holder continuity of weak solutions to the equation under the assumption that coefficients aij (z) belong to the \({VMO_L\cap L^\infty}\) and \({({b_{ij}})_{N \times N}}\) is a constant matrix such that the frozen operator \({L_{z_0}}\) is hypoelliptic.