Some monotonicity results for general systems of nonlinear elliptic PDEs

In this paper we show that minima and stable solutions of a general energy functional of the form $$\int_{\Omega} F(\nabla u,\nabla v,u,v,x)dx$$ enjoy some monotonicity properties, under an assumption on the growth at infinity of the energy. Our results are quite general, and comprise some rigidity results which are known in the literature

Whitham modulation theory for the Kadomtsev-Petviashvili equation

The genus-1 KP-Whitham system is derived for both variants of the Kadomtsev-Petviashvili (KP) equation (namely, the KPI and KPII equations). The basic properties of the KP-Whitham system, including symmetries, exact reductions, and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-deVries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable while all genus-1 solutions of KPII {are linearly stable within the context of Whitham theory.Comment: Significantly revised versio

Exponentially small asymptotic formulas for the length spectrum in some billiard tables

Let $q \ge 3$ be a period. There are at least two $(1,q)$-periodic trajectories inside any smooth strictly convex billiard table, and all of them have the same length when the table is an ellipse or a circle. We quantify the chaotic dynamics of axisymmetric billiard tables close to their borders by studying the asymptotic behavior of the differences of the lengths of their axisymmetric $(1,q)$-periodic trajectories as $q \to +\infty$. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor $q^{-3} e^{-r q}$ times either a constant or an oscillating function, and the exponent $r$ is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the $(1,q)$-periodic trajectories. Our experiments are restricted to some perturbed ellipses and circles, which allows us to compare the numerical results with some analytical predictions obtained by Melnikov methods and also to detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.Comment: 21 pages, 37 figure

Stabilised finite element methods for ill-posed problems with conditional stability

In this paper we discuss the adjoint stabilised finite element method introduced in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on Scientific Computing, and how it may be used for the computation of solutions to problems for which the standard stability theory given by the Lax-Milgram Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to ill-posed problems that have some conditional stability property and prove (conditional) error estimates in an abstract framework. As a model problem we consider the elliptic Cauchy problem and provide a complete numerical analysis for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equation