In this paper we present a formula for the determinant of a matrixvalued elliptic differential operator of even order on a line segment [0, T] with boundary conditions.

A regularized trace formula of first order for the matrix Sturm-Liouville equation with eigenparameter in the boundary conditions is obtained.

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...

the present study is concerned with optimal shape determination of inhomogeneous and temperature dependent domains under steady state heat conduction. such situations are important in many thermal design problems, especially in shape design of electronic components and chips. in the present paper, we formulate the shape optimization problem based on volume minimization of heat conductive materi...

In this paper, we analyze the multi-transmitting formula (MTF) proposed by Liao et al [19]. From the computed reflection coefficients for the fully discrete MTF boundary conditions, we suggest choices for the artificial wave propagation speed, which are different from Liao’s original choice. Theoretical and numerical studies for various incidence angles demonstrate that the suggested choices ef...

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certa...

Using the simple setting of 3D N = 1 supergravity, we show how the tensor calculus of supergravity can be extended to manifolds with boundary. We present an extension of the standard F -density formula which yields supersymmetric bulk-plus-boundary actions. To construct additional separately supersymmetric boundary actions, we decompose bulk supergravity and bulk matter multiplets into co-dimen...

We study the regularity of solutions of the obstacle problem when the obstacle is smooth on each half of the unit ball but only Lipschitz across the shared boundary. We prove that the optimal regularity of these solutions is C 1 2 up to the shared boundary on each half of the unit ball. The proof uses a modification of Almgren’s frequency formula.

In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) ζ-determinant of the Laplacian on a manifold with cylindrical end into the ζ-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula expl...

Using the relation between diffeomorphisms in the bulk and Weyl transformations on the boundary we study the Weyl transformation properties of the bulk metric on shell and of the boundary action. We obtain a universal formula for one of the classes of trace anomalies in any even dimension in terms of the parameters of the gravity action.