We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for stochast...

We prove that the standard conditions that provide unique solvability of a mixed stochastic differential equations also guarantee that its solution possesses finite moments. We also present conditions supplying existence of exponential moments. For a special equation whose coefficients do not satisfy the linear growth condition, we find conditions for integrability of its solution. Keywors. Mix...

The Cauchy problem for 1-dimensional nonlinear sto-chastic partial diierential equations is studied. The uniqueness and existence of solutions in H 2 p (T)-space are proved. 1. Introduction The aim of this paper is to prove the unique solvability of the following one-dimensional nonlinear stochastic partial diierential equa-tions(SPDEs): du = a(t; x; u)u 00 + f(t; x)] dt + k (t)u 0 + g k (t; x)...

Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except a) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient a is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative pr...

We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max-min algebra, which is traditionally defined in terms of unique solvabi...

We construct a perfectly matched absorbing layer for stationary Schrödinger equation with analytic slowly decaying potential in a periodic structure. We prove the unique solvability of the problem with perfectly matched layer of finite length and show that solution to this problem approximates a solution to the original problem with an error that exponentially tends to zero as the length of per...

A language L is the orthogonal concatenation of languages L1 and L2 if every word of L can be written in a unique way as a concatenation of a word in L1 and a word in L2. The notion can be generalized for arbitrary language operations. We consider decidability properties of language orthogonality and the solvability of language equations involving the orthogonal concatenation operation. We esta...

The nonimprovable sufficient conditions for the unique solvability of the problem u ′(t) = `(u)(t) + q(t), u(a) = c, where ` : C(I; ) → L(I; ) is a linear bounded operator, q ∈ L(I; ), c ∈ , are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator ` is not of Volterra’s type with respect to the point a.

The coefficient inverse problem for the degenerate parabolic equation is investigated. minor of this polynomial first power with respect to space variable two unknown time-dependent functions. investigation carried out under given inhomogeneous initial condition, Dirichlet boundary conditions and integral overdetermination conditions. We establish unique solvability named case strong degenerati...