On first and second order planar elliptic equations with degeneracies

Abstract

This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degeneracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.2000 Mathematics Subject Classification. Primary 35J70; Secondaries 35F05, 30G20.The function z ǫ (r, t) = |r| λǫ e it is a first integral of L ǫ in R * × S 1 . That is, L ǫ z ǫ = 0, dz ǫ = 0. Furthermore z ǫ : R + × S 1 −→ C * is a diffeomorphism.

Properties of solutions of a class of planar elliptic operators with degeneracies

“…As was done in [9] and in [10], we establish a one-to-one correspondence between the solutions of the equation Pu = 0 and those of an associated first order equation of type (1.1). To each solution u, we associate its L-gradient w, a solution of an equation (1.1); and vice versa, to each solution w, we can associate its L-potential that satisfies Pu = 0.…”

“…The case when the order of vanishing of L ∧ L is one is considered by the second author in [7], [8], and [10]. The techniques in the above papers will be generalized to the case studied here.…”

Class of hypocomplex structures on the two-dimensional torus

“…where dµ s is the density measure in R 2 . A simple version of this operator was considered in [9] and [10] for other classes of vector fields, and more recently in [5] and [6]. Let T ω g ∈ C α (R), with α = 2 − p − µ p , p = q q − 1 , and µ = σ σ + 1 .…”

Properties of solutions of a class of hypocomplex vector fields

“…That f ∈ L p with p ≤ 2 + σ follows from the fact that Remark 15. Cauchy type integral operators were used in [8] and [9] in connection with other types of vector fields.…”