We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in Rd and 1 < q < ∞, the solvability of the Lq Dirichlet problem for ∆2u = 0 in Ω with boundary data in WA(∂Ω) is equivalent to that of the Lp regularity p...

We report on a comparative study of grating based plasmonic band gap cavities. Numerically, we calculate the quality factors of the cavities based on three types of grating surfaces; uniform, biharmonic and Moiré surfaces. We show that for biharmonic band gap cavities, the radiation loss can be suppressed by removing the additional grating component in the cavity region. Due to the gradual chan...

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)....

We consider the existence, the nonexistence, and the uniqueness of solutions of some special systems of nonlinear elliptic equations with boundary conditions. In a particular case, the system reduces to the homogeneous Dirichlet problem for the biharmonic equation ∆ 2 u = |u| p in a ball with p > 0.

A tensor-product Bézier surface patch x of degree (m,n) is called biharmonic if it satisfies 2x = 0. As shown by Monterde and Ugail (2004), these surface patches are fully determined by their four boundaries. In this note we derive necessary conditions for their existence. © 2006 Elsevier B.V. All rights reserved.