We prove the unique solvability of second order elliptic equations in non-divergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the martingale problem associated with the elliptic equations.

We study the solvability of initial problems for quasilinear equations with Dzhrbashyan–Nersesyan derivative. establish existence a unique solution and apply results to initial-boundary value model systems partial differential equations, in particular, modified system dynamics Scott-Blair medium.

The non-local problem is considered for the partial differential equation of mixed-type with Bessel operator and fractional order. An explicit solution represented by Fourier-Bessel series in given domain. It established connection between data unique solvability problem.

In this work a unique solvability of a class of mixed type partial differential equations with some unbounded coefficients is proved in R2. The estimates of the weighted norms of the solution u and its partial derivatives ux and uy are derived. © 2012 Elsevier Ltd. All rights reserved.

Equations of the form du = (auxixj +Dif i) dt+ ∑ k (σuxi + g k) dwk t are considered for t > 0 and x ∈ R+. The unique solvability of these equations is proved in weighted Sobolev spaces with fractional positive or negative derivatives, summable to the power p ∈ [2,∞).

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3. Mathematics Subject Classification (2010). Primary 20D20; Secondary 20D45.

We study the generalized and regular solvability of first boundary value problem for an odd order equation with changing time direction. Using nonstationary Galerkin method regularization method, existence a solution unique considered are proved. The error estimate is also established.

In this paper we give an elementary proof of the unique, global-in-time solvability of the coagulation-(multiple) fragmentation equation with polynomially bounded fragmentation and particle production rates and a bounded coagulation rate. The proof relies on a new result concerning domain invariance for the fragmentation semigroup which is based on a simple monotonicity argument.

We develop direct and inverse scattering theory for one-dimensional Schrödinger operators with steplike potentials which are asymptotically close to different finite-gap periodic potentials on different half-axes. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.

We investigate the unique solvability of second order parabolic equations in non-divergence form in W 1,2 p ((0, T ) × R), p ≥ 2. The leading coefficients are only measurable in either one spatial variable or time and one spatial variable. In addition, they are VMO (vanishing mean oscillation) with respect to the remaining variables.