We obtain an Einstein metric of constant negative curvature given an arbitrary boundary metric in three dimensions, and a conformally flat one given an arbitrary conformally flat boundary metric in other dimensions. In order to compute the on-shell value of the gravitational action for these solutions, we propose to integrate the radial coordinate from the boundary till a critical value where t...

In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boun...

Recently it has been conjectured that the ground-state of a Markovian Hamiltonian, with one boundary operator, acting in a link pattern space is related to vertically and horizontally symmetric alternating-sign matrices (equivalently fully-packed loop configurations (FPL) on a grid with special boundaries). We extend this conjecture by introducing an arbitrary boundary parameter. We show that t...

We consider the problem of determining the coefficients of a first-order perturbation of the Laplacian in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. From this information we obtain a coupled system of ∂z̄ and ∂z which the coefficients satisfy. As a corollary we show that for a simply connected domain we can determine uniquely the coeffi...

In this paper we study, via variational methods, the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface. The boundary ∂D is assumed to lie within a finite distance of a flat plane and the incident wave is that arising from an inhomogeneous term in the Helmholtz equation whose support lies within some finite distance of the boundary ∂D. Via analysis of an eq...

For p ∈ (1,+∞) and b ∈ (0,+∞] the p-torsion function with Robin boundary conditions associated to an arbitrary open set Ω ⊂ R satisfies formally the equation −∆p = 1 in Ω and |∇u|p−2 ∂u ∂n + b|u| p−2u = 0 on ∂Ω. We obtain bounds of the L∞ norm of u only in terms of the bottom of the spectrum (of the Robin p-Laplacian), b and the dimension of the space in the following two extremal cases: the li...

Super Riemann surfaces of genus 1, with arbitrary spin structures, are shown to be the sets of zeroes of certain polynomial equations in projective superspace. We conjecture that the same is true for arbitrary genus. Properties of superelliptic functions and super theta functions are discussed. The boundary of the genus I super moduli space is determined.

Abstract. The telegraph equation is employed to model wave fields taking into account energy dissipation and media stiffness. The timeharmonic scattered waves generated by a line source incident upon cylindrical obstacles of arbitrary cross-section are studied. Solutions are found to depend strongly on the relative values of the frequency, damping, and stiffness coefficients. These coefficients...

This paper considers the problem of vector tomography on an arbitrary bounded domain in three dimensions. The probe transform of a vector eld is the inner product of the Radon transform of a vector eld with a unit vector, called the probe, which may be a function of the projection orientation. Previous work has given reconstruction formulas for arbitrary elds and for those known to be divergenc...

Motivated by recent developments in string theory, we study the structure of boundary conditions in arbitrary conformal field theories. A boundary condition is specified by two types of data: first, a consistent collection of reflection coefficients for bulk fields on the disk; and second, a choice of an automorphism ω of the fusion rules that preserves conformal weights. Non-trivial automorphi...