In this article we study the solvability of some boundary value problems for inhomogenous biharmobic equations. As a boundary operator we consider the differentiation operator of fractional order in the Miller-Ross sense. This problem is a generalization of the well known Neumann problems.

Abstract. We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Po...

In this paper some existence and nonexistence results for positive solutions are obtained for second-order boundary value problem −u + Mu = f(t, u), t ∈ (0, 1) with Neumann boundary conditions u(0) = u(1) = 0, where M > 0, f ∈ C([0, 1] × R, R). By making use of fixed point index theory in cones, some new results are obtained. Full text

The fundamental boundary value problem in the function theory of several complex variables is the ∂-Neumann problem. The L2 existence theory on bounded pseudoconvex domains and the C∞ regularity of solutions up to the boundary on smooth, bounded, strongly pseudoconvex domains were proved in the 1960s. On the other hand, it was discovered quite recently that global regularity up to the boundary ...

We extend the idea of the Fokas unified transform to investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4 × 4 Lax pair on the half-line. The solution of this system can be expressed in terms of the solution of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be ...

We consider the stability in the inverse problem consisting in the determination of an electric potential q, appearing in a Dirichlet initial-boundary value problem for the wave equation ∂2 t u − ∆u + q(x)u = 0 in an unbounded wave guide Ω = ω × R with ω a bounded smooth domain of R2, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to a wave equat...

It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain Ωh is localized either at the whole lateral surface Γh of the domain, or at a point of Γh, while the eigenfunction decays exponentially inside Ωh. Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and N...

We consider the stability in the inverse problem consisting in the determination of an electric potential q, appearing in a Dirichlet initial-boundary value problem for the wave equation ∂2 t u − ∆u + q(x)u = 0 in an unbounded wave guide Ω = ω × R with ω a bounded smooth domain of R2, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to a wave equat...

We consider a reaction-diffusion equation with Neumann boundary conditions and show that solutions to this problem may be obtained from a problem with periodic boundary conditions and equivariant under O(2) symmetry. We describe the solutions for Hopf bifurcation and mode interactions involving Hopf bifurcation, namely, steadystate/Hopf and Hopf/Hopf. Neumann boundary conditions constrain the s...