in the class of functions u ∈ ⋃T>0W 1,2 p ((−T, 1) × Rd) such that u = 0 for t = 1, where p < d+ 1, q is large enough, ∂t = ∂ ∂t , Di = ∂ ∂xi . A somewhat unusual feature of this problem is that bDiu 6∈ Lp((0, 1)×Rd) for arbitrary u ∈ W 1,2 p ((0, 1) × Rd) even vanishing for t = 1. Therefore, if we solve (1.2) and plug the solution into an equation with different b of the same class, we will generally not obtain a function in Lq even locally. The author is aware of only three similar occasions… Expand

We prove the unique weak solvability and the Feller property for stochastic differential equations with drift in a large class of time-dependent vector fields. This class contains, in particular, the… Expand

We consider a second-order parabolic equation in ℝ d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder… Expand

We consider second-order divergence form uniformly parabolic and elliptic PDEs with bounded and VMOx leading coefficients and possibly linearly growing lower-order coefficients. We look for solutions… Expand

Second-order elliptic equations in $W^{2}_{2}(\mathbb{R}^{d})$ Second-order parabolic equations in $W^{1,k}_{2}(\mathbb{R}^{d+1})$ Some tools from real analysis Basic $\mathcal{L}_{p}$-estimates for… Expand

We prove existence and uniqueness of solutions and Schauder type estimates for a class of elliptic and parbolic equations in with coefficients of the first order derivatives having polynomial growth.… Expand