We study boundary blow-up solutions of semilinear elliptic equations Lu = up + with p > 1, or Lu = e with a > 0, where L is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.

We prove the Kato conjecture for elliptic operators on Rn. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L = −div (A∇) with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate ‖ √ Lf‖2 ∼ ‖∇f‖2.

Elliptic quantum groups can be associated to solutions of the star-triangle relation of statistical mechanics. In this paper, we consider the particular case of the Eτ,η(so3) elliptic quantum group. In the context of algebraic Bethe ansatz, we construct the corresponding Bethe creation operator for the transfer matrix defined in an arbitrary representation of Eτ,η(so3).

Classically, there are many interesting connections between differential operators and the theory of elliptic modular forms and many interesting results have been explored. In particular, it has been known for some time how to obtain an elliptic modular form from the derivatives ofN elliptic modular forms, which has already been studied in detail by R. Rankin in [9] and [10]. When N = 2, as a s...

AILU: A Preconditioner Based on the Analytic Factorization of the Elliptic Operator Martin J. Gander and Frederic Nataf Department of Mathematics, McGill University, Montreal, Canada and CMAP, CNRS UMR7641, Ecole Polytechnique, Palaiseau, France We investigate a new type of preconditioner for large systems of linear equations stemming from the discretization of elliptic symmetric partial differ...

Thermoacoustic tomography is a term for the inverse problem of determining of one of initial conditions of a hyperbolic equation from boundary measurements. In the past publications both stability estimates and convergent numerical methods for this problem were obtained only under some restrictive conditions imposed on the principal part of the elliptic operator. In this paper logarithmic stabi...

were also studied in [47]. It was conjectured in [47] that all these elliptic operators are rigid, generalizing the famous vanishing theorem of Atiyah-Hirzebruch for the Â-genus. There were several rather interesting proofs of these Witten’s conjectures (see [46], [8], [38], [41]). The one relevant to this paper is the proof given in [38], [39] where the main idea was to use the modular invaria...

This paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic Maxwell equations. In particular, a “backslash” multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used. The method of analysis involves comparing the multigrid er...

We present a matrix technique to obtain the spectrum and the analytical index of some elliptic operators defined on compact Riemannian manifolds. The method uses matrix representations of the derivative which yield exact values for the derivative of a trigono-metric polynomial. These matrices can be used to find the exact spectrum of an elliptic operator in particular cases and in general, to g...

Let ⇢ Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by