P.J. McKenna

Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof

Abstract We prove the first genuine “partial differential equation” result on a conjecture concerning the number of solutions of second-order elliptic boundary value problems with a nonlinearity which grows superlinearly at +∞. The proof makes massive use of computer assistance: After approximate solutions have been computed by a numerical mountain pass algorithm, combined with a Newton iteration to improve accuracy, a fixed point argument is used to show the existence of exact solutions close to the approximations.

The study of a nonlinear suspension bridge equation by a variational reduction method

Let L u = utt + uxxxx and H be the complete normed space spanned by the eigenfunction of L. A nonlinear suspension bridge equation (3 < b < 15) in H has at least three solutions. It is shown by a variational reduction method.

A fast algorithm for the solution of the time-independent Gross--Pitaevskii equation

A new efficient numerical method for the solution of the time-independent Gross-Pitaevskii partial differential equation in three spatial variables is introduced. This equation is converted into an equivalent fixed-point form and is discretized using the collocation method at zeros of Legendre polynomials. Numerical comparisons with a state-of-the-art method based on propagating the solution of the time-dependent Gross-Pitaevskii equation in imaginary time are presented.

Global monotone convergence of Newton iteration for a nonlinear eigen-problem

Abstract The nonlinear eigen-problem Ax+F(x)=λx, where A is an n×n irreducible Stieltjes matrix, is considered. Sufficient conditions are given, such that the problem has a unique positive solution and that the Newton iteration for solving this problem converges monotonically. The starting point of the iteration has to be a multiple of the positive eigenvector of A , but it does not need to be close to the solution x .

A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem

A wide variety of articles, starting with the famous paper (Gidas, Ni and Nirenberg in Commun. Math. Phys. 68, 209–243 (1979)), is devoted to the uniqueness question for the semilinear elliptic boundary value problem −Δu=λu+up in Ω, u>0 in Ω, u=0 on ∂Ω, where λ ranges between 0 and the first Dirichlet Laplacian eigenvalue. So far, this question was settled in the case of Ω being a ball and, for more general domains, in the case λ=0. In (McKenna et al. in J. Differ. Equ. 247, 2140–2162 (2009)), we proposed a computer-assisted approach to this uniqueness question, which indeed provided a proof in the case Ω=(0,1)2, and p=2. Due to the high numerical complexity, we were not able in (McKenna et al. in J. Differ. Equ. 247, 2140–2162 (2009)) to treat higher values of p. Here, by a significant reduction of the complexity, we will prove uniqueness for the case p=3.

On the lift-off constant for elastically supported plates

In this paper we continue the study begun by Kawohl and Sweers of the precise constant at which the elastic foundation supporting a bending plate can allow lift-off in the case of downward loading. We provide a number of numerical results and a rigorous result on a different counterexample than the one suggested in Kawohl and Sweers (2002). Important open problems are summarized at the conclusion.

Inclusions and existence proofs for solutions of a nonlinear boundary value problem by spectral numerical methods

For a specific elliptic boundary value problem with quadratic nonlinearity, we give a partial positive answer to an old conjecture concerning the number of solutions. This result is obtained via an existence and enclosure method. For computing the highly accurate solutions needed for this method, a spectral two-grid procedure (combined with a numerical Mountain-Pass algorithm and a Newton iteration) is proposed. Furthermore, Emden’s equation is shown to admit completely spurious approximate solutions which nevertheless have “small” defects—a powerful argument for rigorous enclosure methods.