For a domain R and a Riemannian manifold N R. If u 2 W ( ; N) is an extrinsic (or intrinsic, respectively) biharmonic map. Then u 2 C( ; N). x

In this paper, we establish the existence of at least three solutions to a Navier boundary problem involving the biharmonic equation. The technical approach is mainly base on a three critical points theorem of B. Ricceri. AMS Subject Classifications: 34B15.

We prove that a stationary extrinsic (or intrinsic, respectively) biharmonic map u 2 W ( ; N) from R into a Riemnanian manifold N is smooth away from a closed set of (m 4)-dimensional Hausdor measure zero. x

We study the regularity of solutions to the obstacle problem for the parabolic biharmonic equation. We analyze the problem via an implicit time discretization, and we prove some regularity properties of the solution.

We consider additive Schwarz methods for the biharmonic Dirichlet problem and show that the algorithms have optimal convergence properties for some conforming nite elements. Some multilevel methods are also discussed.

∆p := ∆(|∆u|p−2∆u) is the operator of fourth order, so-called the p-biharmonic (or p-bilaplacian) operator. For p = 2, the linear operator ∆2 = ∆2 = ∆ · ∆ is the iterated Laplacian that to a multiplicative positive constant appears often in the equations of Navier-Stokes as being a viscosity coefficient, and its reciprocal operator noted (∆2)−1 is the celebrated Green’s operator (see [8]). Exis...

New representations of solutions Lamé system with real coefficients via monogenic functions in the biharmonic algebra are found.

Abstract. This paper is concerned with a biharmonic equation under the Navier boundary condition (P∓ε) : ∆u = u n+4 n−4 , u > 0 in Ω and u = ∆u = 0 on ∂Ω, where Ω is a smooth bounded domain in R, n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x0 ∈ Ω a...