Let T denote the unit circle in the complex plane. Given a function f Є Lp (T) , one uses t usual (harmonic) Poisson kernel Ρ (ζ, z) for the unit disk to define the Poisson integral of f , namely h = P[f]. Here we consider the biharmonic Poisson kernel F (ζ, z) for the unit disk to define the notion of F - integral of a given function f Є Lp (T); this associated biharmonic function will be denoted by u = F [f]. We then consider the dilations ur (z) u (rz) r = for z Є T and 0≤r<1. The main result of this paper indicates that the dilations ur are convergent to f in the mean, or in the norm of Lp (T).

This paper is concerned with a version of Saint-Venant type decay estimates for solutions of the Dirichlet problem for the nonlinear biharmonic equation. Energy methods are used to establish a nonlinear integro-differential inequality for a quadratic functional. To this end, integral methods are used to establish exponential decay of solutions for the Dirichlet problem being considered.

This paper deals with the problem of determining of an unknown coefficient in an inverse: elliptic boundary value problem. Using a nonconstant overspecified data, it has been shown that the: solution to this inverse problem exists and is unique.

In this manuscript, we study the Lorentz hypersurfaces of the Lorentz 5-pseudosphere (i.e. the pseudo-Euclidean 5-sphere) $S^5_1$ having three distinct principal curvatures. A well-known conjecture of Bang-Yen Chen on Euclidean spaces says that every submanifold is minimal. We consider an advanced version of the conjecture on Lorentz hypersurfaces of $S^5_1$. We present an affirmative answer to the extended conjecture on Lorentz hypersurfaces with three distinct principal curvatures.

This article addresses the following biharmonic type of the Kirchhoff-Schrödinger-Maxwell system;∆2 w − (a1 +b1∫RN |∇w| 2 )∆w + ηψw = q(w)           in RN,−∆ψ = ηw2                                                                                in RN, (bKSM)in which a1 ,b1 and η are fixed positive numbers and q is a continuous real valued function in R. We are going to prove the existence solution for this system via variational methods, delicate cut-off technique and Pohozaev identity.

1-type and biharmonic curves by using Laplace operator in Lorentzian 3-space are studied and some theorems and characterizations are given for these curves.