We study semilinear evolution equations [Formula: see text] posed on a Hilbert space [Formula: see text], where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from [Formula: see text] to itself, and [Formula: see text], [Formula: see text], [Formula: see text]. In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation wi...

Abstract. This paper is companion to our earlier work [8] (see also announcement [7]). Let M be a closed manifold and Y be an embedded hypersurface, such that there is a decomposition of M = M1 ∪M2 into two manifolds with boundary M1 and M2 , with M1 ∩M2 = Y . In [8] we proved the decomposition formula for detζ∆ the ζ-determinant of a Dirac Laplacian ∆ on M . The contributions coming from M1 an...

The relationship between the boundaries for Hill and Lagrange stability in orbital element space is modified in the case of resonantly interacting planets. Hill stability requires the ordering of the planets to remain constant while Lagrange stability also requires all planets to remain bound to the central star. The Hill stability boundary is defined analytically, but no equations exist to def...

We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C-graphs. In our case, the same monotonicity formula does not apply in the same way. In th...

We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (≡ APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative index theorem for the difference of two APS-index classes associated to differe...

I offer a simple and useful formula for the resolvent of a small rank perturbation of large matrices. I discuss applications of this formula, in particular, to analytical and numerical solving of difference boundary value problems. I present examples connected with such problems for the difference Laplacian and estimate numerical efficiency of the corresponding algorithms.

We give a smooth analogue of the classical Schläfli formula, relating the variation of the volume bounded by a hypersurface moving in a general Einstein manifold and the integral of the variation of the mean curvature. We extend it to variations of the metric in a Riemannian Einstein manifold with boundary, and apply it to Einstein cone-manifolds, to isometric deformations of Euclidean hypersur...

In this paper, we study Vanishing Mean Oscillation vector fields on a compact manifold with boundary. Inspired by the work of Brezis and Niremberg, we construct a topological invariant — the index — for such fields, and establish the analogue of Morse’s formula. As a consequence, we characterize the set of boundary data which can be extended to nowhere vanishing VMO vector fields. Finally, we s...

We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary v...

This lecture note covers topics on boundary conformal field theory, modular transformations and the Verlinde formula, and boundary logarithmic CFT. An introductory review on CFT with boundary and a discussion of its applications to logarithmic cases are given. LCFT at c = −2 is mainly discussed. This article is based on the lectures at the International Summer School on Logarithmic Conformal Fi...