The solvability in $W^{2}_{p}(bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This
generalizes to a very large extent the case of equations with continuous or VMO coefficients.
We extend several known results on solvability in the Sobolev spaces $W^{1}_{p}$, $pin[2,infty)$, of SPDEs in divergence form in $bR^{d}_{+}$ to equations having coefficients which are discontinuous in the space variable.
We present several results on solvability in Sobolev spaces $W^{1}_{p}$ of SPDEs in divergence form in the whole space.
We consider a second-order parabolic equation in $bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Holder continuous in the space variables. We show that g
lobal Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.
We prove It^os formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.
We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to a similar
problem for smooth functions with values in a polyhedron.
