Enclosure for the Biharmonic Equation

In this paper we give an enclosure for the solution of the biharmonic problem and also for its gradient and Laplacian in the $L_2$-norm, respectively

Reconstructing obstacles using CGO solutions for the biharmonic equation

In this article, we study an inverse problem for detecting unknown obstacle by the enclosure method using the Dirichlet to Neumann map as measurements. We justify the method for the impenetrable obstacle case involving the biharmonic equation. We use complex geometrical optics solutions with logarithmic phase to reconstruct some non-convex part of the obstacle. The proof is based on the global $L^p$-estimates for the gradient and Laplacian of the solutions of the biharmonic equation for $p$ near $2$

Asymptotic First Eigenvalue Estimates for the Biharmonic Operator on a Rectangle

We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small.Comment: 27 pages, 4 diagrams, 2 table

Computer-assisted enclosures for fourth order elliptic equations

We describe a computer-assisted method for proving existence and multiplicity of solutions of fourth order nonlinear elliptic boundary value problems: we compute a good numerical approximation of a solution and certain defect bounds with computer-assistance, and then obtain a rigorous proof of the existence of an exact solution close to the numerical one by a fixed-point argument

05391 Abstracts Collection -- Algebraic and Numerical Algorithms and Computer-assisted Proofs

From 25.09.05 to 30.09.05, the Dagstuhl Seminar 05391 ``Algebraic and Numerical Algorithms and Computer-assisted Proofs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. Links to extended abstracts or full papers are provided, if available