We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue. We address the problem of minimizing this eigenvalue among suitable classes of domains. We prove the existence of an optimal convex domain of fixed measure. Mathematics Subject Classification (2000)....

The lack of a general maximum principle for biharmonic equations suggests to study under which boundary conditions the positivity preserving property holds. We show that this property holds in general domains for suitable linear combinations of Dirichlet and Navier boundary conditions. The spectrum of this operator exhibits some unexpected features: radial data may generate nonradial solutions....

This paper deals with the shape preserving interpolation problem for visualization of 3D positive data. A required display of 3D data looks smooth and pleasant. A rational bi-cubic function involving six shape parameters is presented for this objective which is an extension of piecewise rational function in the form of cubic/quadratic involving three shape parameters. Simple data dependent cons...

We consider a class of time dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of problems covers important cases such as Fokker-Planck type equations and aggregation models, which have been studied intensively in the past decades. In this pape...

Tile role of reconstruction in avoiding oscillations in upwind sc|_emes is reexamined, with the aim of providing simple, concise proofs. In one dimension, it is shown that if the reconstruction is any arbitrary function bounded by neighboring ('ell averages and increasing within a ('ell for increasing data, the resulting scheme is monotonicity preserving, even though the reconstructed function ...

We develop a family of second-order implicit-explicit (IMEX) schemes for the stiff BGK kinetic equation. The method is asymptotic-preserving (can capture the Euler limit without numerically resolving the small Knudsen number) as well as positivity-preserving — a feature that is not possessed by any of the existing second or high order IMEX schemes. The method is based on the usual IMEX Runge-Ku...

In this paper, we present high order accurate positivity-preserving conservative remapping algorithm which is based on the multi-resolution weighted essentially non-oscillatory (WENO) reconstruction. We use a third-order method 2D quadrilateral meshes as an example to algorithm. The can effectively remap physical variables after mesh rezoning in ALE By calculating intersection exactly, does not...

We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we pr...