Related papers: The Prolongation Problem for the Heavenly Equation

In this article we discuss the solvability of some class of fully nonlinear equations, and equations with p-Laplacian in more general conditions by using a new approach given in [1] for studying the nonlinear continuous operator. Moreover…

In this paper we show how a second order scalar uniformly elliptic equation on divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary…

Consider an operator equation (*) $B(u)+\ep u=0$ in a real Hilbert space, where $\ep>0$ is a small constant. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following…

We give existence and regularity results for solutions of some nonlinear elliptic problems. The equations we deal with are modeled on a problem which involves in its principal part an anisotropic operator, a Hardy-type potential, and a…

In this paper we consider a quantum open system and treat the master equation with some restricted dissipator which consists of a set of projection operators (projectors). The exact solution is given under the commutable approximation (in…

We investigate properties of exponential operators preserving the particle number using combinatorial methods developed in order to solve the boson normal ordering problem. In particular, we apply generalized Dobinski relations and methods…

We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions.

The aim of the present paper is to give extensions of the cosine-sine functional equation.

In this paper, we determine necessary and sufficient conditions for the generalized Bessel function to be in certain subclasses of starlike and convex functions. Also, we obtain several corollaries as special cases of the main results,…

We prove existence, uniqueness, and regularity of viscosity solutions to the stationary and evolution obstacle problems defined by a class of nonlocal operators that are not stable-like and may have supercritical drift. We give sufficient…

We propose a flexible approach for computing the resolvent of the sum of weakly monotone operators in real Hilbert spaces. This relies on splitting methods where strong convergence is guaranteed. We also prove linear convergence under…

An uniform expansion of the Legendre functions of large indices are considered by using the WKB approach. We obtain the recurrent formula for the coefficients of uniform expansion and compare them with the uniform expansion of the Bessel…

A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic…

In this paper we consider the evolution equation $\partial_t u=\Delta_\mu u+f$ and the corresponding Cauchy problem, where $\Delta_\mu$ represents the Bessel operator $\partial_x^2+(\frac{1}{4}-\mu^2)x^{-2}$, for every $\mu>-1$. We…

Highly oscillatory integrals, such as those involving Bessel functions, are best evaluated analytically as much as possible, as numerical errors can be difficult to control. We investigate indefinite integrals involving monomials in $x$…

We study the Hessian of the fundamental solution to the parabolic problem for weighted Schr\"odinger operators of the form $\frac 12 \Delta+\nabla h-V$ proving a second order Feynman-Kac formula and obtaining Hessian estimates. For…

We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…

We consider the parabolic Lam\'{e} system on a bounded domain. We focus on two types of inequalities for higher-order derivatives of solutions. The first is related to an $L^p$-$L^p$ estimate locally in time in the Lebesgue space setting,…

The linearized Vlasov equation for a plasma system in a uniform magnetic field and the corresponding linear Vlasov operator are studied. The spectrum and the corresponding eigenfunctions of the Vlasov operator are found. The spectrum of…

Using Jacobi's identity we derive a simple expression for the Bessel functions of integer order in terms of combinations of powers and hyperbolic functions of the same argument.