Interior \(W^{1,p}\) regularity and Hölder continuity of weak solutions to a class of divergence Kolmogorov equations with discontinuous coefficients (Q383904)

In this paper, the authors consider a class of Kolmogorov equation in divergence form defined in a bounded open domain of \(\mathbf R^{N+1}\). Equations of this form have applications in the theory of stochastic processes, in the kinetic theory and in mathematical finance. They have been widely studied by Lanconelli and his school. In this paper, the authors assume the principal part is symmetric, bounded, uniformly elliptic and belongs to \(\mathrm{VMO}_L\). Furthermore the drift term is assumed constant and such that the frozen operator is hypoelliptic. Under such assumptions, the authors prove interior \(W^{1,p}\) \((1 <p< \infty)\) regularity and Hölder continuity of weak solutions to the equation.