Some relations that ensure the parabolicity of the type

This paper investigates sufficient conditions for the parabolicity type of the domain R 2 ∖ K, where K is a compact set, with respect to a general variational functional IΦ. A known criterion for the conformal parabolicity of a Riemannian metric requires that the coordinate functions be harmonic. We significantly generalize this result by establishing new differential, rather than modul-capacitary, conditions for Φ-parabolicity at infinity. The work introduces and studies a special class of differential 1-forms, Λ qc x2 (D), which generate quasiconformal mappings used to construct appropriate mapping functions. The main results, formulated as Theorems 1 and 2, provide verifiable criteria involving the interplay between the functional Φ, a form Ψ of parabolic type, and auxiliary differential forms θ and ω. These criteria are expressed via the essential boundedness of certain quantities, such as Vθ,Φ,Ψ, and differential inequalities involving the Hodge operator. The proofs leverage techniques from quasiconformal mapping theory, potential theory (including Perron’s method), and the calculus of variations. A key corollary generalizes the harmonic coordinate condition to the case of quadratic functionals associated with uniformly elliptic operators in divergence form. The obtained conditions are shown to be checkable in specific model situations.