Topics on euclidean classical field equations with unique vacuua

Abstract

We discuss the real classical field equation (-δ+μ²)φ+λF(φ) = f, where μ²>o, λ≥o, F ∈ c^∞(R), and aF (a)≥0 for all a ∈ R, and where the source function f belongs to various function spaces contained in the Sobolev space H₋₁ (R ^d). We review a number of results whose proofs are to appear elsewhere, on existence of a solution φ ∈ H_l(R ^d), the correspondences between the function spaces of f and φ (regularity properties), contractivity properties and uniqueness of φ, and analytic dependence on λ and functional differentiability in f. We mention some of the ideas from the proofs, including a few alternate methods we have not discussed elsewhere. We prove a new result on positivity preservingness: if f is a nonnegative measure in H₋₁(R ^d), then 0 ≤ φ ∈ H₁(R ^d), a result that is also valid for μ² = 0. Finally, we give an intuitive interpretation of some of the results for the case of d = 1 dimension.

Work supported in part by the NSF under Grant No. MCS 7701748 2