The purpose of this paper is to establish the regularity the weak solutions for the nonlinear biharmonic equation { ∆2u + a(x)u = g(x, u), u ∈ H2(RN ), where the condition u ∈ H2(RN ) plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.

We study the following two nonlinear evolution equations with a fourth order (biharmonic) leading term: −∆2u− 1 22 (|u|2 − 1)u = ut in Ω ⊂ R or R and −∆2u + 1 22 ∇ · ((|∇u|2 − 1)∇u) = ut in Ω ⊂ R or R with an initial value and a Dirichlet boundary conditions. We show the existence and uniqueness of the weak solutions of these two equations. For any t ∈ [0,+∞), we prove that both solutions are i...

In a radar imaging problem using broad-band, low-frequency waves, we encounter the problem of solving Poisson's equation over a very large rectangular grid, typically ve thousand times thousand pixels. In addition, no information about boundary values is available. In order to select suitable solutions we solve the Poisson equation under the side condition that some criterion function, usually ...

This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation. The weak Galerkin method, first introduced by two of the authors (J. Wang and X. Ye) in [52] for second order elliptic problems, is based on the concept of discrete weak gradients. The method uses completely discrete finite element functions and, using certain discrete spaces an...

We study existence and positivity properties for solutions of Cauchy problems for both linear and semilinear parabolic equations with the biharmonic operator as elliptic principal part. The self-similar kernel of the parabolic operator ∂t + ∆ 2 is a sign changing function and the solution of the evolution problem with a positive initial datum may display almost instantaneous change of sign. We ...