Generalized solutions of boundary-value problems for differential equations of general form

Abstract

We describe a new class of instanton effects in string compactifications that preserve only N = 1 supersymmetry in four dimensions. As is well-known, worldsheet or brane instantons in such a background can sometimes contribute to an effective superpotential for the moduli of the compactification. We generalize this phenomenon by showing that such instantons can also contribute to new multi-fermion and higher-derivative F -terms in the low-energy effective action. We consider in most detail the example of heterotic compactification on a Calabi-Yau threefold X with gauge bundle V , in which case we study worldsheet instanton effects that deform the complex structure of the moduli space associated to X and V . We also give new, slightly more economical derivations of some previous results about worldsheet instantons in Type IIA Calabi-Yau compactifications.

On Dirichlet, Poncelet and Abel problems

“…Note also that some explicit necessary and sufficient conditions of uniqueness breakdown of solution of the Dirichlet problem (and some others boundary value problems) for partial differential equations with constant coefficients were obtained earlier for an arbitrary ellipse (see e.g., [16] and [17]). Answers in those works were formulated in the form of condition (1.5), which was to be a hint in our present investigations.…”

“…17) where H(x, y) is a Hamilton function of the system. Obviously H(x, y) is an in-tegral of the system (3.17), i.e.…”

“…Note also that the case of an ellipse was considered in works by A. Huber [32], R. Alexandrjan [4], V. I. Arnold [6]. For information on small smoothness and on more general equations, see the book [17]. Let Ω be an arbitrary bounded domain which is convex with respect to characteristics of the equation (1.1), i.e.…”

Boundary properties of solutions of differential equations and general boundary-value problems

“…which we shall need below [10,47]. By a generalized solution of the Dirichlet problem for the equation (4.5) with righthand side f ∈ D (L 0 ) we mean an element u ∈ D(L 0 ) satisfying the following "integral" identity for any element v ∈ H l 0 with minimal operator L 0 :…”

Dirichlet and Neumann Problems for String Equation, Poncelet Problem and Pell-Abel Equation

“…Besides, note the Dirichlet problem (2) has some nontrivial solution u in Sobolev spaces iff the Neumann problem u ν * | C = 0 with the conormal ν * for the equation (1) has a nonconstant solution u [3].…”