Given a prescribed boundary of a Bézier surface, we compare the Bézier surfaces generated by two different methods, i.e., the Bézier surface minimising the biharmonic functional and the unique Bézier surface solution of the biharmonic equation with prescribed boundary. Although often the two types of surfaces look visually the same, we show that they are indeed different. In this paper, we prov...

We study the behavior of two maps in an effort to better understand the stability of ωlimit sets ω(x, f ) as we perturb either x or f , or both. The first map is the set-valued function Λ taking f in C(I ,I) to its collection of ω-limit points Λ( f ) = ⋃x∈I ω(x, f ), and the second is the map Ω taking f in C(I ,I) to its collection of ω-limit sets Ω( f )= {ω(x, f ) : x ∈ I}.We characterize thos...

In this note, we characterize the $f$-harmonic maps and bi-$f$-harmonic with potential. We prove that every map potential from complete Riemannian manifold, satisfying some conditions is a More, study case of conformal between equidimensional manifolds.

The discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh-refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it. One approach for this is the careful design of an intermediate triangulation with one-level refinements and with the remaining difficulty to design some interpolati...

The paper studies boundary integral operators of the bi{Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving mixed boundary value problems for the biharmonic equation on a nonsmooth plane domain, analyse the solvability of the co...

The study of biharmonic functions under the ordinary (Euclidean) Laplace operator on the open unit disk D in C arises in connection with plate theory, and in particular, with the biharmonic Green functions which measure, subject to various boundary conditions, the deflection at one point due to a load placed at another point. A homogeneous tree T is widely considered as a discrete analogue of t...

Biharmonic equation is one of the most existing operator in many areas, especially in fluid and solid mechanics, but difficult to solve due to the fourth order derivatives in the differential equation. In this paper we deal with the use of the local particular solution method to solve the two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmoni...

Surface-enhanced Raman Scattering (SERS) of rhodamine 6G (R6G) adsorbed on biharmonic metallic grating structures was studied. Biharmonic metallic gratings include two different grating components, one acting as a coupler to excite surface plasmon polaritons (SPP), and the other forming a plasmonic band gap for the propagating SPPs. In the vicinity of the band edges, localized surface plasmons ...

The main result of this paper is the construction of a conformally covariant operator in two dimensions acting on scalar fields and containing fourth order derivatives. In this way it is possible to derive a class of Lagrangians invariant under conformal transformations. They define conformal field theories satisfying equations of the biharmonic type. Two kinds of these biharmonic field theorie...